SYNTHETIC APERTURE IMAGING OF DIRECTION AND FREQUENCY DEPENDENT REFLECTIVITIES

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1 SYNTHETIC APERTURE IMAGING OF DIRECTION AND FREQUENCY DEPENDENT REFLECTIVITIES LILIANA BORCEA, MIGUEL MOSCOSO, GEORGE PAPANICOLAOU, AND CHRYSOULA TSOGKA Abstrat. We introdue a syntheti aperture imaging framework that takes into onsideration diretional dependene of the refletivity that is to be imaged, as well as its frequeny dependene. We use an l minimization approah that is oordinated with data segmentation so as to fuse information from multiple sub-apertures and frequeny sub-bands. We analyze this approah from first priniples and assess its performane with numerial simulations in an X-band radar regime. Key words. syntheti aperture imaging, refletivity, minimal support optimization.. Introdution. We introdue and analyze a novel algorithm for syntheti aperture radar (SAR) imaging, where a moving reeive-transmit platform probes a remote region with signals f(t) and reords the sattered waves. The platform spans a large syntheti aperture so that high resolution images of the region may be obtained by proessing the reorded data. A related appliation is inverse syntheti aperture radar (ISAR), where the reeive-transmit antenna is stationary, and the syntheti aperture is due to the motion of an unknown satterer. If this motion is known or an be estimated, the problem an be restated mathematially as SAR imaging of the satterer, using the referene frame that moves with it. A shemati of the SAR imaging setup is in Figure.. The reordings u(s, t) at the moving reeive-transmit platform depend on two time variables: the slow time s and the fast time t. The slow time parametrizes the trajetory of the platform, and it is disretized in uniform steps h s, alled the pulse repetition rate. At time s the platform is at loation r(s). It emits the signal f(t) and reeives the baksattered returns u(s, t). The fast time t runs between onseutive signal emissions t (, h s ), and we assume a separation of time sales: The duration of f(t) is smaller than the round trip travel time of the waves between the sensor and the imaging region, and the latter is smaller than h s. In the usual syntheti aperture image formulation the refletivity is modeled as a two dimensional funtion of loation y on a surfae of known topography, say flat for simpliity. The assumption is that eah point on the surfae reflets the waves the same way in all diretions, independent of the diretion and frequeny of the inident waves. This simplifies the imaging proess and makes the inverse problem formally determined: the data are two-dimensional and so is the unknown refletivity funtion. The refletivity an be reonstruted by the reverse time migration formula,,, 7] I( y) = j dt u(s j, t)f ( t τ(s j, y) ). (.) Department of Mathematis, University of Mihigan, Ann Arbor, MI 89 borea@umih.edu Gregorio Millán Institute, Universidad Carlos III De Madrid, Madrid 89, Spain mososo@math.um.edu Stanford Mathematis Department, Serra Mall Bldg. 8, Stanford CA 9 papanio@math.stanford.edu Mathematis and Applied Mathematis, University of Crete and IACM/FORTH, GR-79 Heraklion, Greee tsogka@uo.gr

2 r(s) f(t) y Fig... Setup for imaging with a syntheti aperture. Here s j are the slow time emission-reording instants, spaed by h s, and the image is formed by superposing over the platform trajetory the data u(s j, t), mathfiltered with the time reversed emitted signal f(t), delayed by the roundtrip travel time τ(s j, y) = r(s j ) y / between the platform loation r(s j ) and the imaging point y. The bar denotes omplex onjugate and is the wave speed in the medium whih is assumed homogeneous. The assumption of an isotropi refletivity may not always be justified in appliations. Baksatter refletivities are in general funtions of five variables: the loation y on the known (flat) surfae, the two angles of inidene and the frequeny. Thus, the inverse problem is underdetermined and we annot expet a reonstrution of the five dimensional refletivity with a migration approah. Diret appliation of (.) will produe low-resolution images of some effetive, position-dependent refletivity, and there will be no information about the diretivity and frequeny dependene of the atual refletivity. The reonstrution of frequeny dependent refletivities with syntheti aperture radar has been onsidered in 8], where Doppler effets are shown to be useful in inversion, and in, ], where data are segmented over frequeny sub-bands, and then images are formed separately, for eah data subset. Data segmentation is a natural idea, and we show here how to use it for reonstruting both frequeny and diretion dependent refletivities. The main result in this paper is the introdution and analysis of an algorithm for imaging diretion and frequeny dependent refletivities of strong, loalized satterers. This algorithm is based on l optimization. It reonstruts refletivities of loalized satterers by seeking among all those that fit the data model the ones with minimal spatial support. Array imaging algorithms based on l optimization are proposed and analyzed in, 8,,,,, ]. They onsider only isotropi, frequeny independent

3 refletivities. A diret extension of l optimization methods to imaging diretion and frequeny dependent refletivities amounts to solving a grand optimization problem for a very long vetor ρ of unknowns, the disretized refletivity over spatial loations on the imaging grid, the angles of inidene/baksatter and the frequeny. It has onsiderable omputational omplexity beause of the high dimension of the spae in whih the disretized refletivity vetor lies. It also does not take into aount the fat that many unknowns are tied to the same spatial loation points within the disretized image window. The syntheti aperture imaging algorithm introdued in this paper is designed to reonstrut effiiently diretion and frequeny dependent refletivities by ombining two main ideas: The first is to divide the data over arefully alibrated sub-apertures and frequeny sub-bands, and solve an l optimization problem to estimate the refletivity for eah data subset. Data segmentation is useful assuming that the refletivity hanges ontinuously with the diretion of probing and the frequeny, so that we an approximate it by a pieewise onstant funtion, pointwise in the imaging window. Over a sub-aperture of small enough linear size a, the platform reeives sattered waves from a narrow one with opening angle of the order a/l, where L is the distane from the platform to the imaging window, and we an approximate the refletivity by that at the enter angle. Similarly, we an approximate the refletivity by a onstant over a small enough frequeny sub-band. Then, we an use l optimization to estimate the refletivity as a funtion of loation for eah data subset. The size of the sub-apertures and sub-bands determine the resolution of the reonstrution. The larger they are, the better the expeted spatial resolution of the refletivity. But the resolution is worse over diretion and frequeny dependene. The alibration of the data segmentation over sub-apertures and sub-bands reflets this trade-off. The seond idea ombines the l optimizations by seeking refletivities that have ommon spatial support. Instead of a single vetor ρ, the unknown is a matrix with olumns of spatially disretized refletivities. Eah olumn orresponds to a diretion of probing from a sub-aperture and a entral frequeny in a sub-band. The values of the entries in the olumns are different, but they are zero (negligible) in the same rows. Moreover, the forward model, whih is derived here from first priniples, maps eah olumn of the refletivity matrix to the entries in the data subsets via one ommon refletivity-to-data model matrix. The optimization an then be arried out within the multiple measurement vetor (MMV) formalism desribed in, 9,, ]. The MMV formalism is used for solving matrix-matrix equations for an unknown matrix variable whose olumns share the same support but have possibly different nonzero values. We show in this paper how to redue the syntheti aperture imaging problem to an MMV format. The olumns of the unknown matrix are assoiated with the disretized spatial refletivities for different diretions and frequenies. The solution of the MMV problem an be obtained with a matrix (,)-norm minimization where one seeks to minimize the l norm of the vetor formed by the l norms of the rows of the unknown refletivity matrix. The solutions obtained this way preserve the ommon support of the olumns of the unknown matrix. This paper is organized as follows. We begin in setion with the formulation of the imaging problem. We derive the data model, desribe the omplexity of the inverse problem, and motivate our imaging approah. The foundation of this approah is in setion, where we show how to redue the imaging problem to an MMV format. The imaging algorithm is desribed in setion and its performane is assessed with

4 numerial simulations in setion. The presentation in setions - uses the so-alled start stop approximation, whih neglets the motion of the reeive-transmit platform over the duration of the fast time data reording window. This is for simpliity and also beause the approximation holds in the X-band radar regime used in the numerial simulations. However, the imaging algorithm an inlude Doppler effets due to the motion of the reeive-transmit platform, as explained in setion. We end with a summary in setion 7.. Formulation of the imaging problem. The data model is desribed in setion.. Then, we review briefly imaging of isotropi refletivity funtions via migration and l optimization in setion.. The formulation of the problem for diretion and frequeny dependent refletivities is in setion... Syntheti aperture data model. In syntheti aperture imaging we usually assume that the data u(s, t), depending on the slow time s and the fast time t, an be modeled with the single sattering approximation. For an isotropi and frequeny independent refletivity funtion ρ = ρ( y) we have dω u(s, t) = π û(s, ω)e iωt, (.) with Fourier transform û(s, ω) given by û(s, ω) k f(ω) Ω d y ρ( y) exp iωτ(s, y) ] (π r(s) y ). (.) Here k = ω/ is the wavenumber and the integral is over points y in Ω, the support of ρ. The model (.) uses the so-alled start-stop approximation, where the platform is assumed stationary over the duration of the fast time reording window. We use this approximation throughout most of the paper for simpliity, and beause it holds in the X-band radar regime onsidered in the numerial simulations. However, the results extend to other regimes, where Doppler effets may be important, as explained in setion. The inverse problem is to invert relation (.) and thus estimate ρ( y), given u(s j, t) at the slow time samples s j = (j )h s, for j =,..., N s. Here h s is the slow time sample spaing. The inversion is usually done by disretizing (.), to obtain a linear system of equations for the unknown vetor ρ of disretized refletivities. The support Ω in (.) is not known, so the inversion is done in a bounded searh domain Y on the imaging surfae, assumed flat. We all Y the image window. The reonstrution of ρ in Y is a solution of the linear system, as we review briefly in setion.. The disretization of Y is adjusted so that it is ommensurate with the expeted resolution of the image in range and ross-range. The range diretion is the projetion on the imaging plane of the unit vetor pointing from the imaging loation y Y to the platform loation. The ross-range diretion is orthogonal to range. It is well known in imaging that the range resolution is determined by the auray of travel time estimation, whih in turn is determined by the temporal support of f(t). Thus, it is useful to have a short pulse f(t) whose support is of order /B, where B is the bandwidth. The range resolution with suh pulses is of order /B. The ross-range resolution is proportional to the entral wavelength, whih is why the emitted signals are typially modulated by high arrier frequenies ω o /(π). If L is a typial distane between the platform and the imaging window and A is the length of the flight path,

5 so that the platform reeives waves within a one of opening angle A/L, the rossrange resolution is of the order λ o L/A, where λ o = π/ω o is the arrier wavelength. We assume that ω o B, whih is usually the ase in radar. In syntheti aperture imaging appliations like SAR, the platform emits relatively long signals f(t) so as to arry suffiient energy to generate strong satter returns, and thus high signal to noise ratios. Examples of suh signals are hirps, whose frequeny hanges over time in an interval entered at the arrier frequeny ω o /(π). To improve the preision of travel time estimation, and therefore range resolution, the returns u(s j, t) are ompressed in time via math-filtering with the time reversed emitted signal ]. Moreover, to remove the large phases and therefore avoid unneessarily high sampling rates for the returns, the data are migrated via travel time delays alulated with respet to a referene point y o in the imaging window. The ombination of these two data pre-proessing steps is alled down-ramping. For the purposes of this paper it suffies to assume that f(t) is a linear hirp, in whih ase the Fourier transform f(ω) of the ompressed signal has approximately the simple form f(ω) f(ω o ) ωo πb,ω o+πb](ω), (.) where ω,ω ](ω) denotes the indiator funtion of the frequeny interval ω, ω ]. The down-ramped returns are ( ) ] dω dt u s, t t + τ(s, y o ) f( t ) = π f(ω)û(s, ω)e iω t+τ(s, y o), (.) and we let d be the vetor of the samples of its Fourier transform d = (d(s j, ω l )) j=,...ns,l=,...,n ω, d(s, ω) = f(ω)û(s, ω)e iωτ(s, yo). (.) The size of the vetor d is N s N ω. The linear relation between the unknown refletivity vetor ρ and the downramped data vetor d follows from (.) and (.). We write it as Aρ = d, (.) where the entries in ρ C Q are proportional to ρ( y q ), with y q the Q disretization points of the image window Y, and with the onstant of proportionality taken to be the area of a grid ell. The refletivity ρ is mapped by the refletivity-to-data matrix A C N SN ω Q to the data d. The assumption of frequeny independent refletivity leads to a set of deoupled systems of equations A(ω l )ρ = d(ω l ) indexed by the frequeny ω l, where the entries of the N s Q matries A(ω l ) are A j,q (ω l ) = kl f(ω l ) ] (π r(s j ) y q ) eiω l τ(s j, y q) τ(s j, y o). (.7) Here k l = ω l /, l =,..., N ω, j =,..., N s, and q =,..., Q... Imaging isotropi refletivities. Imaging of the isotropi refletivities amounts to inverting the linear system (.). When this system is underdetermined, there are two frequently used hoies for piking a solution: either minimize the Eulidian norm of ρ or its l norm. The first hoie gives ρ = A d, (.8)

6 where A is the pseudo-inverse of A. If A is full row rank, A = A (AA ). The inversion formula (.8) also applies to overdetermined problems, where ρ is the least squares solution and A = (A A) A, for full olumn rank A. The hoie of the imaging window Y and its disretization is an essential part of the imaging proess and, depending on the objetives and available prior information, we may be able to ontrol whether the system (.) is overdetermined or not. We explain in Appendix B that by disretizing Y in steps ommensurate with expeted resolution limits we an make the olumns of A nearly orthogonal. This means that in the overdetermined ase A A is lose to a diagonal matrix. We also shown in Appendix B that in the underdetermined ase, for oarse enough sampling of the slow time s and frequeny ω, the rows of A are nearly orthogonal, and therefore AA is lose to a diagonal matrix. Thus, in both ases, A is approximately A up to multipliative fators, and we an therefore image the support of ρ with A d. This is in fat the migration formula (.) written in the Fourier domain, up to a geometrial fator, sine the amplitude in (.7) is approximately onstant for platform trajetories that are shorter than the imaging distane and for bandwidths B ω o. If we know that the imaging sene onsists of a few strong, loalized satterers, as we assume here, a better estimate of ρ is given by the optimization min ρ suh that Aρ d ɛ. (.9) Here ɛ is an error tolerane, ommensurate with the noise level in the data, and and are the l and the Eulidian norm, respetively. We refer to, 8,,, ] for studies of imaging with l optimization. The main result in this ontext is that when there is no noise so that ɛ =, the refletivities are reovered exatly provided that the inner produts of the normalized olumns of A are suffiiently small. An extension of the optimization to nonlinear data models that aount for multiple sattering effets in Y, is onsidered in ]. A resolution study of imaging with l optimization is in ]... Imaging diretion and frequeny dependent refletivities. In general, baksatter refletivities are funtions of five variables: the loation y Y, the unit diretion vetor m and the frequeny ω. Hene, ρ = ρ( y, m, ω). (.) This means that the down-ramped data model is more ompliated than assumed in equations (.) and (.) or, equivalently, after disretization, in (.)-(.7). In integral form it is given by d(s, ω) = f(ω)û(s, ω)e iωτ(s, yo) = k f(ω) exp iω τ(s, y) τ(s, y o ) ]] d y ρ( y, m(s, y), ω) ( ), (.) π r(s) y Ω where m(s, y) is the unit vetor pointing from the platform loation r(s) to y in the image window Y. In disretized form we still have a linear system like (.), exept that now ρ is a vetor of QN s N ω unknowns, the disretized values of ρ in the image window Y. Extending the inversion approahes desribed in the previous setion to this model means inverting approximately the matrix A with a very large number of olumns.

7 r(s α ) t α m α y o Fig... Shemati of the geometry for one sub-aperture entered at the loation r(s q ) of the reeive-transmit platform. We annot expet the migration formula (.) to give an aurate estimate of the refletivity as a funtion of five variables, as pointed out in the introdution. The l optimization approah works, but it beomes impratial for the large number QN s N ω of unknowns. Moreover, it does not take into aount the fat that the entries in ρ indexed by the slow time and frequeny pairs (j, l), with j =,..., N s and l =,..., N ω, refer to the same loations y q on the imaging grid. The imaging approah introdued in this paper gives an effiient way of estimating diretion and frequeny dependent refletivity funtions of strong loalized satterers in Y. It uses an approximation of the model (.), motivated by the expetation that the baksatter refletivity should not hange dramatially from one platform loation to the next and from one frequeny to another. Instead of disretizing ρ over all five variables at one, we disretize it only with respet to the loation in the image window Y, for one probing diretion and frequeny at a time. To do so, we separate the data over subsets defined by arefully alibrated sub-apertures and sub-bands, and freeze the diretion and frequeny dependene of the refletivity for eah subset. The grand optimization is divided this way into smaller optimizations for Q unknowns, whih are then oupled by requiring that the unknown vetors share the same spatial support in the imaging window Y.. Redution to the Multiple Measurement Vetor framework. We present here an analysis of how we an write the linear relation between the diretion and frequeny dependent refletivity and the data as a linear matrix system AX = D, (.) where the unknown is the matrix X with Q rows. The entries in the rows orrespond to the disretization of this refletivity at the Q grid points y q in Y. Eah olumn of 7

8 X depends on the refletivity at the baksattered diretion defined by the enter of a sub-aperture and the enter frequeny of a sub-band. The data are segmented over N α sub-apertures and N β sub-bands and are grouped in the matrix D. The objetive of this setion is to desribe the data segmentation and derive the linear system (.), whih an be inverted with the MMV approah as explained in setion. We begin in setion. with a single sub-aperture and sub-band. We show in Lemma. that with proper alibration of the sub-aperture and sub-band size, the refletivity-to-data matrix has a simple approximate form. Its entries have nearly onstant amplitudes while the phases depend linearly on the slow time and frequeny parametrizing the data subset. This simplifiation allows us to transform the linear system via oordinate rotation to a referene one, for all data subsets, as shown in setion.. The matrix A in (.) orresponds to the referene sub-aperture and sub-band, and the statement of the result is in Proposition.... The sub-aperture and sub-band segmentation. We enumerate the sub-apertures by α =,..., N α, and denote by s α the slow time that orresponds to their enter loation r(s α). The hoie of the sub-aperture size a is important, and we address it in the next setion. For now it suffies to say that it is small enough so that we an approximate it by a line segment, as illustrated in Figure.. The unit tangent vetor along the trajetory, at the enter of the sub-aperture, is denoted by t α, and the platform motion will be assumed uniform, at speed V t α. The unit vetor from the referene loation y o in the image window to r(s α) is m α. We all it the range vetor for the α sub-aperture. The range (distane) to the imaging window is = r(s α) y o. (.) Eah sub-aperture is parametrized by the slow time offset from s α, denoted by s = s s α a V, a ]. (.) V We do not index it by α beause it belongs to the same interval for eah sub-aperture. The disretization of s is at the slow time sample spaing h s, and there are n s = a V h s + sample points, where a/(v h s ) is rounded to an integer. Similarly, we divide the bandwidth in N β sub-bands of support b B, entered at ωβ, and let ω be the frequeny offset ] ω = ω ωβ πb, πb. (.) We sample the sub-band with n ω points. The refletivity dependene on the diretion and frequeny is denoted by the supersript pair (α, β), and by disretizing it with the Q points in Y we obtain the vetor of unknowns ρ (α,β) C Q. It is mapped to the data vetor d (α,β) with entries given by the samples of d(s α + s, ωβ + ω). The mapping is via the n sn ω Q refletivity-to-data matrix A (α,β) desribed in Lemma.... Refletivity-to-data model for a single sub-aperture and sub-band. Here we explain how we an hoose the size of the sub-apertures and frequeny subbands so that we an simplify the refletivity-to-data matrix. The alibration depends 8

9 on the size of the imaging window Y, whih is quantified with two length sales and Y α = max ( y q y o ) m α, (.) q=,...q Y α = max q=,...q P α( y q y o ). (.) Here P α = I m α m T α is the projetion on the ross-range plane orthogonal to m α, and I is the identity matrix. The length sale Y α gives the size of Y viewed from the range diretion m α, and Yα is the ross-range size. The first onstraints on the aperture a and the ross-range size Yα of the imaging window state that they are not too small, and thus imaging with adequate resolution an be done with the data subset. Expliitly, we ask that for all α =,..., N α, a λ o ay α λ o (Y α ) λ o. (.7) The inequalities on the left involve two Fresnel numbers a /(λ o ) and (Y α ) /(λ o ), whose magnitudes define the imaging regime. If these numbers were small, we would be in a Fraunhofer diffration regime, with approximately planar wavefronts on the sale of the sub-aperture and of the size of the imaging window. We onsider a Fresnel diffration regime, where these numbers are larger and we an get better resolution of images. The ross-range resolution is λ o /a, and naturally, the middle inequality in (.7) says that the image window is larger than the resolution limit. In the range diretion we suppose that Y α b λ o, (.8) where /b is the range resolution for the sub-bands, and we used that b B ω o. While we would like to have a and b large so as to get good spatial resolution of the unknown refletivity, we reall that ρ is frozen in our disretization in the small frequeny sub-band and in the narrow one of opening angle of the order a/, with axis defined by the enter r(s α) of the sub-aperture and the referene point y o. The larger a and b are, the oarser the estimation of the diretion and frequeny dependene of ρ. The more rapid the variation of ρ with diretion and frequeny, the smaller a and b should be to represent it, at the expense of resolution. There is also a trade-off between resolution and the omplexity of the inversion algorithm. By onstraining a and b so that and b ω o Yα, (.9) λ o /a a Y α λ o L, α a Y α λ o L α, (.) we an simplify the mapping between the refletivity and the data subset, as stated in Lemma.. This simplifiation allows us to use the effiient MMV framework to solve the large optimization problem for the entire data set, by onsidering jointly the 9

10 smaller problems for the segmented data in an automati way. The key observation here is that the unknown refletivities for eah data subset share the same spatial support. This is what the MMV formalism is designed to apture. The next lemma gives the form of the refletivity-to-data matrix in the linear system A (α,β) ρ α,β = d (α,β), (.) for the (α, β) data subset. It is an approximation of the system (.) restrited to the rows indexed by the n s slow times in the α aperture and the n ω frequenies in the β band. The expression of A (α,β) is derived in appendix A. Lemma.. Under the assumptions (.7)-(.), and with the pulse model (.), the matrix A (α,β) onsists of n ω bloks A (α,β) ( ω l ) indexed by the frequeny offset ω l, for l =,..., n ω. Eah blok is an n s Q matrix with entries defined by A (α,β) j,q ( ω l ) = k o f(ω o ) (π ) exp { i(k β + ω l /) m α y q ik β V s j t α P α y q where k β = ω β /, and y q = y q y o. y q P α y } q + ik β, (.).. Multiple sub-aperture and sub-band model as an MMV system. It remains to show how to write equations (.) in the matrix form (.) with a refletivity-to-data matrix independent of the sub-apertures and sub-bands. This is aomplished via a rotation, that brings all the sub-apertures to a single referene sub-aperture. But to do this, we need to know that eah data subset has a similar view of the image window. Mathematially, this is expressed by the following two additional onstraints on a and b max α N α, q Q b ( mα m ) y q, (.) and max α N α, β N β, q Q ( akβ t α P α ak ) t P y q, (.) L The onstraint (.) states that the imaging points remain within the range resolution limit b/ for all the apertures. The onstraint (.) states that the imaging points remain within the ross-range resolution limits, as well. The derivation of the linear system (.) is in appendix A and the result is stated in the next proposition. Proposition.. Under the same assumption as in Lemma. and in addition, supposing that onditions (.) and (.) hold, we an ombine the linear systems (.) in the matrix equation (.). The referene sub-aperture and sub-band are indexed by α = and β =. The unknown matrix X has Q rows and N α N β olumns indexed by (α, β). Its entries are X (α,β) q = ρ (α,β) y q P α y ] q q exp ik β m α y q + ik β, (.)

11 where ρ (α,β) q = ρ( y q, m α, ω β), m α = r(s α) y o r(s α) y o. (.) The data matrix D has n s n ω rows and N α N β olumns indexed by (α, β). We organize the equations in bloks indexed by the frequeny ω l, for l =,..., n ω. The entries of D are defined in terms of the down-ramped data vetors d (α,β) as where we reall that D (α,β) j ( ω l ) = (π) k o f(ω o ) d(α,β) ( s j, ω l ), (.7) d (α,β) ( s j, ω l ) = d ( s α + s j, ω β + ω l ), (.8) and d(s, ω) is defined in (.). The refletivity to data matrix A has n ω bloks indexed by ω l, denoted by A( ω l ). Eah blok is an n s Q matrix with entries A j,q ( ω l ) = exp i ω l V s ] j m y q ik t P y q. (.9) L Note that the produt of the refletivity-to-data matrix A with eah olumn of X an be interpreted, up to a onstant multipliative fator, as a Fourier transform with respet to the range offset m y and ross-range offset t P y in Y. Equation (.) shows that the olumns of X differ from eah other by a linear phase fator in y, whih amounts to a rotation of the oordinate system of the α sub-aperture, and a quadrati fator whih orrets for Fresnel diffration effets. Thus, the linear system (.) gives roughly the Fourier transform of the refletivity ρ for different range diretion views, and the imaging problem is to invert it to estimate ρ.. Inversion algorithm. Here we desribe the algorithm that estimates the refletivity by inverting the linear system (.). By onstrution, the olumns of the Q N α N β unknown matrix X have the same spatial support, beause they represent the same spatial refletivity funtion. Thus, we formulate the inversion as a ommon support reovery problem for unknown matries with relatively few nonzero rows 9,, 9, ]. This Multiple Measurement Vetor (MMV) formulation has been studied in,, 9] and has been used suessfully for soure loalization with passive arrays of sensors in ] and for imaging strong sattering senes, where multiple sattering effets annot be negleted, in ]. In the MMV framework the support of the unknown matrix X is quantified by the number of nonzero rows, that is the row-wise l norm of X. If we define the set rowsupp(x) = {q =,..., Q s.t. e T q X l }, (.) where e T q X is the q th row of X and e q is the vetor with entry in the q th row and zeros elsewhere, then the row-wise l norm of X is the ardinality of rowsupp(x), Ξ (X) = rowsupp(x). To estimate X we must to solve the optimization problem min Ξ (X) s.t. AX = D, (.)

12 but this is an NP hard problem. We solve instead the onvex problem min J, (X) s.t. AX = D, (.) whih gives, under ertain onditions on the model matrix A, ], the same solution as (.). In (.) J, denotes the (, )-norm J, (X) = m e T q X l, (.) q= whih is the l norm of the vetor formed by the l norms of the rows of X. Furthermore, beause data are noisy in pratie, we replae the equality onstraint in (.) by AX D F < ɛ, where F is the Frobenius norm and ɛ is a tolerane ommensurate with the noise level of the data. There are different algorithms for solving (.) or its reformulation for noisy data. We use an extension of an iterative shrinkage-thresholding algorithm, alled GeLMA, proposed in 7] for matrix-vetor equations. This algorithm is very effiient for solving l -minimization problems, and has the advantage that the solution does not depend on the regularization parameter used to promote minimal support solutions, see 7] for details. Algorithm GeLMA-MMV Require: Set X =, Z =, and pik the step size µ and the regularization parameter γ. repeat Compute the residual E = D AX X X + µa (Z + E) e T q X sign( e T q X l µγ) et q X l µγ e T q X l e T q X, q =,..., Q Z Z + γe until Convergene After estimating X with Algorithm, we reover the disretized diretion and frequeny dependent refletivity using equation (.), ρ (α,β) q = X q (α,β) y q P α y ] q exp ik β m α y q ik β, (.) for the imaging points y q = y o + y q indexed by q =,..., Q, the sub-apertures indexed by α =,..., N α and frequeny sub-bands indexed by β =,..., N β.. Numerial simulations. We begin in setion. with the numerial setup, whih is in the regime of the GOTCHA Volumetri data set ] for X-band persistent surveillane SAR. Then we present in setions. and. the simulation results... Imaging in the X-band (GOTCHA) SAR regime. The numerial simulations generate the data with the model (.), for various sattering senes. The regime of parameters is that of the GOTCHA data set, where the platform trajetory is irular at height H = 7.km, with radius R = 7.km and speed V = 7m/s. The signal f(t) is sent every.m along the trajetory, whih gives a slow time spaing h s =.s. The arrier frequeny is ω o /(π) = 9.GHz and the bandwidth is B = MHz. The waves propagate at eletromagneti speed = 8 m/s, so the

13 wavelength is λ o =.m. The image window Y is at the ground level, below the enter of the flight trajetory, and the distane from the platform to its enter y o is L =.8km. It is a square, with side length Y = Y of the order of m. The size of the sub-apertures is a = m and the width of eah sub-band is b = B/. Given these parameters, the nominal resolution limits are λ o L/a = 7.m, /b = 7.m. The image window Y is disretized in uniform steps h = m in range and h = m in ross-range, and the refletivity is modeled as pieewise onstant on the imaging grid. The image disretization affets the quality of the reonstrution with l optimization. It must be oarse enough so that uniqueness of the l minimizer holds, and yet fine enough so that modeling errors due to off-grid plaement of the unknown are ontrolled. We refer to ] for a study of this trade-off. To illustrate the performane of the algorithm, we present in the next two setions results for various imaging senes onsisting of small satterers supported on one pixel of the imaging grid, or over multiple adjaent pixels. The latter is for representing larger satterers for whih the diretion dependent refletivity an be motivated by Snell s law of refletion at their surfae. The results presented in the next setions ompare the images obtained with reverse time migration and the algorithm proposed in this paper, hereby referred to as the MMV algorithm. The migration image is omputed with the formula (π) I( y) = ko f(ω o ) n s n ω hh n s n ω j= l= d(s j, ω l ) r(s j ) y e iω l τ(s j, y) τ(s j, y o) ], (.) whih is a weighted version of (.), where the weights are hosen so as to provide a quantitative estimate of the unknown ρ. That is to say, when we substitute the data model in (.), under the assumption of an isotropi and frequeny independent refletivity we get that I( y) peaks at the true loation of the satterers and its value at the peaks equals the true refletivity there. Let us verify the assumptions (.7)-(.) with the GOTCHA parameters. The Fresnel numbers are larger than one, as stated in (.7), a λ o L =. and (Y ) λ o L =.. The size of the imaging region and the range resolution satisfy (.8). Moreover, b ω o Y λ o L/a =., whih is onsistent with (.9), and (.) is satisfied as well, a Y λ o L = a Y λ o L =.... Single frequeny results. We begin with imaging results at the arrier frequeny, where we assume we know the range of the satterers and seek to reonstrut their refletivity as a funtion of ross-range and diretion. The image window

14 KM true MMV... 8 Fig... Estimation of an isotropi, frequeny independent refletivity as a funtion of rossrange, using N α = 8 onseutive, non-overlapping apertures. The exat refletivity is shown with the full green line, the migration result with the blue line and the MMV inversion result with the broken line. The absissa is ross-range in meters. extends over m in ross-range, and it is sampled in steps h = m, where we reall that λ o L/a = 7.m. The first result displayed in Figure. is for an isotropi, frequeny independent refletivity of satterers, N α = 8 onseutive, non-overlapping apertures and noiseless data. We display in green the true refletivity, in blue the refletivity estimated with formula (.), and with broken line the result of the MMV inversion algorithm. In the legend we abbreviate the migration formula result with the letters KM, standing for Kirhhoff Migration. The figure shows that the MMV algorithm reonstruts exatly the refletivity, and that the weighted migration formula (.) does indeed give quantitative estimates of the refletivity. However, the migration estimates deteriorate when the refletivity is anisotropi and frequeny dependent, as illustrated next. The results displayed in Figure. are obtained with N α = onseutive, nonoverlapping apertures. The refletivity depends on two variables: the ross-range loation and the sattering diretion, parameterized by the slow time s α, for α =,...,. In disretized form it gives a matrix R true with row index orresponding to the pixel loation in the image window, and olumn index orresponding to the subaperture. The reonstrution of this matrix is denoted by R. The green and broken lines in the top plots in the figure display the true and reonstruted refletivity at the peak diretion, vs. ross-range. Expliitly, for eah pixel in the image i.e., eah row q in R true or R, we display the maximal entry. The migration image of the refletors is independent of the diretion and is plotted with the blue line. The results show that we have small satterers, whih are well estimated by the MMV algorithm even for noisy data. The migration method identifies orretly the loations of the satterers, but the refletivity value is no longer aurate beause only a few sub-apertures see

15 8.9 KM true MMV.9 KM true MMV satterer. satterer. satterer Fig... Estimation of the refletivity as a funtion of diretion and ross-range loation for a sene with satterers. The top plots show the refletivity as a funtion of ross-range (the absissa in meters), for the peak diretions. The left plot is for noiseless data and the right plot is for data ontaminated with % additive noise. The green line is the exat peak value and the broken line the one obtained with MMV. The blue line is obtained with migration. The bottom plots display the refletivity of eah satterer as a funtion of sub-aperture i.e., the slow time index α =,...,, where is the number of sub-apertures. The left plot is for the true refletivity, the middle plot is for the noiseless reonstrution and the right plot is for the noisy reonstrution. eah refletor, as we infer from the bottom plots desribed next. This also implies a deterioration in the ross-range resolution whih is more visible in the next set of results in Figure.. Naturally, the migration image gives no information about the diretion dependene of the refletivity. In the bottom plots in Figure. we show the value of the refletivity of eah satterer as a funtion of diretion, parameterized by the slow time s α. That is to say, we identify first the row indexes q in R true or R at whih we have a strong satterer (see top plots) and then display those rows. The left plot is for the true refletivity, the middle is for the noiseless reonstrution, and the right is for the noisy reonstrution. We observe that the MMV method reonstruts the diretion dependent refletivity exatly in the noiseless ase, and very well in the noisy ase. In Figure. we illustrate the effet of the anisotropy of the refletivity on the imaging proess. We display the results the same way as in in the previous figure. The point is to notie that while the MMV method estimates aurately the diretion dependent refletivity in all ases, the migration method performs poorly when the anisotropy is strong, meaning that eah satterer is seen only by one sub-aperture at a time (top plots). The resolution is not that orresponding to the atual aperture of a = m, but that for a single sub-aperture of a = m. The middle and bottom row plots show how migration images improve when the anisotropy of the refletivity is weaker and more sub-apertures see eah satterer... Multiple frequeny results. Now we onsider multiple frequeny sub-

16 8.9 KM true MMV satterer 8... satterer KM true MMV satterer 8... satterer KM true MMV satterer 8... satterer Fig... Imaging results for anisotropi refletivities, N α = sub-apertures and data ontaminated with % additive noise. From top to bottom we derease the anisotropy. This an be seen from the right olumn plots whih show the refletivity of eah satterer for eah sub-aperture. The middle olumn shows the reonstruted refletivity as a funtion of diretion with the MMV algorithm. bands and thus seek to estimate the refletivity as a funtion of range, ross-range, diretion and frequeny. We have N ω sub-bands of width b, and we sample eah of them at n ω = frequenies. The number of sub-apertures is N α = 8. The imaging region is a square of side m and it is sampled in ross-range in steps h = m and in range in steps h = m. We denote, as before, by R true the true matrix of disretized refletivities and by R the reonstruted ones. These are matries of size Q N α N ω and we display them in the image window Y as follows: For eah pixel in the image window i.e., a row q in R true or R, we display the maximum entry, the peak value of the refletivity at point y q over diretions and frequenies. One we identify the loation of the satterers from these images, i.e., determine their assoiated rows, we display the entries in these rows, to illustrate the diretion and frequeny dependene of their refletivity. These are the middle and right plots in the figures. We begin in Figure. with a single frequeny sub-band (N ω = ), N α = 8 onseutive, non-overlapping sub-apertures and data ontaminated with % additive noise. The anisotropi refletivity model has four satterers, as illustrated in the top

17 range satterer ross range range.8. range... satterer ross range ross range Fig... Results for a single frequeny sub-band and N α = 8 onseutive, non-overlapping subapertures. On the top we show the true refletivity as a funtion of loation (middle) and diretion (right). On the bottom we show the reonstruted refletivity with % additive noise. Left plot is the migration image, middle plot is the MMV image and the right plot is the diretional dependene of the refletivity reonstruted with the MMV algorithm. The axes in the left images are ross-range and range in meters. The absissa in the right plots are sub-aperture index α =,..., N α and the ordinate is the index of the satterer (from to ). plots. Eah satterer is seen by a single sub-aperture. The reonstruted refletivity is shown in the bottom plots. On the left we show the migration image, whih is blurry and is unable to loate the weaker satterers. The MMV algorithm gives an exellent reonstrution as shown in the middle and right plots. The results in Figures. and. are for N ω = 8 onseutive, non-overlapping frequeny bands and N α = 8 onseutive, non-overlapping sub-apertures. The differene between the figures is the strength of the satterers and their anisotropy. The results in Figure. show that the MMV algorithm reonstruts well the loation of the satterers and the diretion dependene of their refletivity. The frequeny dependene of the weaker satterers is not that aurate, likely beause the bandwidth is small and all frequenies are similar to the arrier. As in Figure., the migration image is blurrier and does not loate the weak satterers. Figure. shows that the migration image improves when all satterers are of approximately the same strength and they have weaker anisotropy. The last illustration onsiders a larger satterer with diretion dependent refletivity, supported over four adjaent pixels, and a small isotropi satterer with frequeny dependent refletivity. The data is ontaminated with % additive noise. We note that the migration method gives the orret loation of the large satterer, but not the value of its refletivity. Moreover, it gives a blurry image of the small satterer. The MMV algorithm determines well the support of both satterers, as well as aurate estimates of the refletivity as a funtion of diretion and frequeny. 7

18 range.. satterer.... satterer ross range sub band range satterer satterer ross range sub band x range 8 ross range Fig... Results with N ω = 8 frequeny intervals, N α = 8 apertures and data ontaminated with % noise. On the top we show the true refletivity as a funtion of loation (left) and diretion (middle), and frequeny (right). In the middle row we show the reonstruted refletivity with the MMV algorithm. The plot in the bottom row is the migration image. The axes in the left images are ross-range and range in meters. The absissa in the middle and right plots are the sub-aperture and sub-band index and the ordinate is the index of the satterer (from to ).. Doppler effets. All the results up to now use the start-stop approximation of the data model, whih neglets the motion of the platform over the fast time reording window. Here we extend them to regimes where Doppler effets are important. We begin in setion. with the derivation of the generalized data model that inludes Doppler effets, and an assessment of the validity of the start-stop approximation. Then we explain in setion. how to inorporate these effets in our imaging algorithm... Data model with Doppler effets. For simpliity we first derive the data model for an isotropi refletivity ρ = ρ( y). Then we extend it in the obvious way to diretion and frequeny dependent refletivities in a sub-aperture indexed by α and sub-band indexed by β, with refletivity ρ (α,β) ( y). 8

19 range satterer satterer ross range sub band range satterer satterer ross range sub band... range.... ross range Fig... Results with N ω = 8 frequeny intervals, N α = 8 apertures and data ontaminated with % noise. On the top we show the true refletivity as a funtion of loation (left), diretion (middle), and frequeny (right). In the middle row we show the reonstruted refletivity with the MMV algorithm. The plot in the bottom row is the migration image. The axes in the left images are ross-range and range in meters. The absissa in the middle and right plots are the sub-aperture and sub-band index and the ordinate is the index of the satterer. The sattered wave u(s, t) reorded at the transmit-reeive platform is given by u(s, t) = d y ρ( y) Ω = d yρ( y) Ω t t dt dt f (t )G(t t, r(s + t ), y)g(t t, y, r(s + t)), f ( t (t) ) (π) r ( s + t (t) ) y r(s + t) y (.) where t (t) is the solution of the equation t + r(s + t ) y = t 9 r(s + t) y, (.)

20 range... satterer... satterer ross range sub band range... satterer... satterer ross range sub band... range.... ross range Fig..7. Results with N ω = 8 frequeny intervals, N α = 8 apertures and data ontaminated with % noise. On the top we show the true refletivity as a funtion of loation (left), diretion (middle), and frequeny (right). The refletivity is supported in pixels, four of them are adjaent and represent an extended satterer, whih is frequeny independent. The other pixel supports a small satterer with frequeny dependent refletivity. In the middle row we show the reonstruted refletivity with the MMV algorithm. The plot in the bottom row is the migration image. The axes in the left images are ross-range and range in meters. The absissa in the middle and right plots are the sub-aperture and sub-band index and the ordinate is the index of the pixel in the support of the satterer. and we used the expression of the Green s funtion of the wave equation G(t, r, y) = δ t r y / ] π r y, and the single sattering approximation. The expression (.) is simply the spherial wave emitted from r(s + t ), over the duration t of the pulse, sattered isotropially at y, and then reorded at r(t + s). Up to the single sattering approximation, this is an exat formula. Expanding with respet to t the arguments in (.) and (.) we

21 obtain u(s, t) = d yρ( y) Ω (π r(s) y ) ( + O(V t/l) ) f ( ( ( V V t )) ) ( ( V t γ(s, y) + O τ(s, y) / + γ(s, y) + O R where we introdued the Doppler fator γ defined by γ(s, y) = r (s) m(s, y), m(s, y) = V t ))], (.) R r(s) y r(s) r(y). (.) We assume that the platform is moving at onstant speed V along a trajetory with unit tangent denoted by t(s), and with radius of urvature R assumed omparable to the range L. Thus, ( V ) γ(s, y) = O, beause the platform speed is typially muh smaller than, the wave speed, and we an neglet the residual in (.) whih is even smaller than γ, beause over the duration of the fast time window the platform travels a small distane ompared with the radius of urvature V t R L. We have thus the data model u(s, t) f t ( γ(s, y) ) τ(s, y) ( γ(s, y) )] d yρ( y) (π r(s) y ), (.) Ω whih inludes first order Doppler effets. The start-stop approximation is valid when the Doppler fator in the argument of f in (.) is negligible. Although γ is small, f osillates at the arrier frequeny ω o whih is large and, depending on the sale of the fast time t, the Doppler fator may play a role. Reall that t is limited by the slow time spaing h s. In pratie the duration of the fast time window may be muh smaller than h s, although it must be large enough so that the platform an reeive the ehoes delayed by the travel time, τ(s, y). Expliitly, t = O(L/) + O(/B), where L/ is the sale of the travel time and /B is the sale of the duration of the signal. We onlude that the start stop approximation holds when ( ωo L ω o tγ(s, y) = O V ) + O ( ωo B V ). In the GOTCHA regime, onsidered in the numerial simulations in setion, we have ω o L V =.9, ω o V B =., so ω o tγ(s, y) is slightly less than one. We may inlude it in the data model, but it amounts to a onstant additive phase that has no effet in imaging. To see this, let us take the Fourier transform with respet to t in (.) û(s, ω) k d yρ( y) f ω ( + γ(s, y) )] exp iω ( + γ(s, y) ) τ(s, y) ] (π r(s) y ), (.) Ω

22 and expand the arguments over the slow time s and imaging point y. We use the approximation r (s) V t(s ) n(s ) V s ], (.7) R where s is the slow time offset from the enter s of the aperture, and t(s ) is the unit tangent to the trajetory of the platform at the enter point. The seond term in (.7) aounts for the urved platform trajetory, with unit vetor n(s ) orthogonal to t, in the plane defined by t and the enter of urvature, and R the radius of urvature. We also have and ( V s ) m(s, y) m(s, y o ) = O + O L ( Y ω o τ(s, y) = ω o τ(s, y o ) + O(k o V s) + O(k o y). Substituting in (.) and using the parameters of the GOTCHA regime, we see that, ωγ(s, y)τ(s, y) ω o γ(s, y ) oτ(s, y o ), so indeed, the Doppler effet amounts to a onstant phase term... Imaging algorithm with Doppler effets. The model of the downramped data with the Doppler orretion follows from (.), d(s, ω) = f ω ( + γ(s, y o ) )] û(s, ω) exp iω ( + γ(s, y o ) ) τ(s, y o ) ] k f ω ( + γ(s, yo ) )] Ω d y f ω ( + γ(s, y) )] ρ( y) exp iω ( + γ(s, y) ) τ(s, y) iω ( + γ(s, y o ) ) τ(s, y o ) ] ( π r(s) y ). (.8) L ), We are interested in diretion and frequeny dependent refletivities, so to use formula (.8), we onsider next the α th sub-aperture and the β th sub-band, where we an replae ρ by ρ (α,β) ( y). The data is denoted by d (α,β) ( s, ω), where s = s s α and ω = ω ωβ. The goal of the setion is to inlude Doppler effets in the statements of Lemma. and Proposition., whih are the basis of our imaging algorithm. We begin with the observation that ωγ(s, y)τ(s, y) = ω r (s) ( r(s) y) = ωγ(s, y o )τ(s, y o ) ω r (s) where y = y y o, and r (s) is given by (.7), and assume heneforth that V Yα y, (.9) b ω o. (.) This is onsistent with our previous assumptions beause Yα and V, and allows us to approximate the Doppler fator in the argument of the Fourier transform