Reduction to monostatic focusing of bistatic ormotion uncompensated SAR surveys

A. Monti Guarnieri and F. Rocca

Abstract: The authors study the response caused by a point scatterer in the general case of bistaticsurveys. The relative time–space function, called the flat top hyperbola, is derived. The complexityof this curve stimulates the reduction of a general bistatic survey to a monostatic one, following thetechniques used in seismic data processing. The scatterers corresponding to a single received pulsein a bistatic configuration are located on the intersection of the ground plane with an isochronal ellipsoidthat has source and receiver as foci. These scatterers are then considered as targets of a monostatic con-figuration. Thus, a bistatic configuration is reduced to a monostatic one by identifying an approximatespace-varying transfer function between the two configurations. This paper, a sequel to a recently pub-lished paper, also addresses the problem of motion compensation, in that the first survey could also bemonostatic, and the location of the second monostatic survey is at choice. The use of a digital elevationmodel is shown to be essential for the correct determination of the transfer function in the general case.It is not necessary if the source and receivers have a constant distance and follow the same trajectory.

1 Introduction

In these last years, there has been an impressive growth inthe interest in bistatic radar [1]. In the field of syntheticaperture radar (SAR), constellations of multiple spacebornesensors, like the Cartwheel, should be available in the nearfuture [2, 3]; the first bistatic airborne SAR acquisitionshave already been successfully completed [4–6]. As a conse-quence, a growing literature is appearing on bistatic focusing.Depending on the geometry, the bistatic configuration, theantenna aperture and the bandwidth, the processing of suchsystems can be as simple as the tuning of a monostatic pro-cessor or a real challenge [7]. Several papers approach the‘stationary’ case, with parallel orbits and same velocity,either by exploiting proper approximations, like smallantenna apertures [8, 9], or by exact solutions [10, 11].Other papers approach the more general case of differentorbits and velocities, apart from the exact but computationallyexpensive backprojection technique by Ding and Munson[12], and this case is taken yet again to a monostatic approachby a proper, approximated pre-processing [7, 13]. Despite thisrich literature, the development of an efficient phase pre-serving focusing scheme is still a challenging issue becauseof the requirement for massive data processing on one sideand the design of wide-aperture, wide-bandwidth SAR onthe other.

This paper, read in part at the 2004 EUSAR meeting,comes after [11], where introductory remarks on the tech-nique of transforming bistatic into monostatic surveyswere discussed. For a proper understanding of the geophysi-cal approach, the terminology and the general modelling, werefer the reader to D’Aria et al. [11], Deregowski and Rocca

[14] and Fomel [15]. With respect to D’Aria et al. [11],where the constant offset case between source and receiverwas discussed, there is a significant innovation in that the3D character that is peculiar to more general bistatic SARsurveys is taken into account.

We remind the reader that the technique that we discusshere is more usual in geophysics where up to 6000 receiverscan operate simultaneously and where, normally, thesurveys have a full 3D character. We refer to the recentwork by Fomel [15] for a complete and thorough discussionof the approximations implicit in this approach. Theapproximation is weaker for non-uniform media where thevelocity changes are significant in the volume shared bythe two acquisitions. This situation is, in general, far fromthose encountered with SAR unless localised atmosphericor ionospheric artifacts change significantly the velocityof the rays of one survey with respect to the other.

The aim of this work is to establish a general frameworkto categorise the bistatic surveys, transforming them into amonostatic one. Bistatic surveys may use opportunity trans-mitters, and thus the surveying geometry might be ratherunusual. Therefore it is useful to find a general tool(space variant, indeed) that applies to all bistatic geome-tries, naturally space variant, transforming them into thespace invariant monostatic geometry, indicating theinstances where the transformation proposed in D’Ariaet al. [11] could not be applied. Sure enough, in manycases, extreme squints, and unlikely geometries can beavoided. However, there will also be cases where the oppor-tunity source and the position of the receiver will be suchthat this work could be of interest. Thus, we decided notto enter into the discussion of synthetic cases that wouldanyway be biased towards a specific geometry and notnecessarily the one of interest in the case to be studied.

We begin with a short discussion of the problem of focus-ing, in the case of general bistatic surveys, and we providethe travel-time curve [called the flat top hyperbola (FTH)]that gives, for a pre-assigned position of a point scattererlocated on the ground plane, the total travel time fromsource to the scatterer and then back to the receiver asa function of the positions of the source and receiver.

# The Institution of Engineering and Technology 2006

IEE Proceedings online no. 20045105

doi:10.1049/ip-rsn:20045105

Paper first received 8th October 2004 and in final revised form 8th November2005

The authors are with the Dipartimento di Elettronica, Politecnico di Milano,Piazza Leonardo da Vinci 32, 20133 Milano, Italy

E-mail: monti@elet.polimi.it

IEE Proc.-Radar Sonar Navig., Vol. 153, No. 3, June 2006 199

Then, focusing can be carried out as usual by combining(say with a weighted sum) the received data at the times cor-responding to the FTH, so that we can estimate the reflectiv-ity of the scatterer. This technique could be computationallyexpensive. Furthermore, it does not fully characterise theinherent differences between bistatic and monostaticconfigurations.

Successively, we introduce a different approach to focus-ing. We identify first an ellipse on the ground plane and itsintersection with the isochronal ellipsoid that has as foci thesource and the receiver. One single-received pulse in thebistatic survey is what we would have seen if we had scat-terers nowhere else but along that ellipse. The illuminationfunction provided by the directivity of the transmitter and ofthe receiver obviously would limit this ellipse to a smallpart. Then, we can carry out a monostatic survey onto thiselliptical distribution of scatterers; further, we evaluatethe outcome of such a survey. So, we will be able to carryout a transformation from any received pulse in a generalbistatic configuration into a travel time curve (the general-ised smile) in a monostatic one; that is, we will identifythe transfer function between the two configurations. Thepeculiarities of the generalised smile are useful to appreci-ate the impact of the bistatic configuration. Finally, we willdeal with the case of motion compensation where both con-figurations are monostatic but different.

2 Bistatic geometry

Before discussing the general geometry of a bistatic survey, letus first appreciate its differences with the monostatic case. In amonostatic situation with straight orbit, the acquisition hasrotational symmetry – the axis being the orbit itself – andthe survey may be considered 2D, rather than 3D. This com-paction from a 3D space to a 2D one cannot happen forcurved orbits or in a general bistatic survey where sourceand receiver follow different paths. Thus, the non-stationarityof the 3D geometry along azimuth has to be characterised,necessarily complexifying any focusing technique.

Let us consider a general bistatic geometry as in Fig. 1,where the target P is imaged by the bistatic system com-posed of the sensors S1 and S2:

† v1 and v2 are the velocities of the two sensors on theirtrajectory, assumed rectilinear;

† rz1 and rz2 are the closest approach distances of the targetwith respect to the two sensors;† tz1 and tz2 are the azimuth times at which each sensor isin the closest approach with respect to the target.

We can assume S1 to be transmitting and S2 receiving,and define t as the azimuth, slow-time. The sensors–target-distances at any time t are then

r1ðt;PÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt� tz1Þ

2v21 þ r2

z1

qð1Þ

r2ðt;PÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit2v2

2 þ r2z2

qð2Þ

where we assumed, without loss of generality, that theorigin for t-axis is such that tz2 ¼ 0. The bistatic range isobtained by adding (1) and (2)

rb ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt� tz1Þ

2v21 þ r2

z1

qþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit2v2

2 þ r2z2

qð3Þ

fb ¼2p

lrb ð4Þ

The expression (3), usually defined as ‘double square root’(DSR) summation, describes the bistatic curve known asthe FTH. The bistatic Doppler phase history (DPH), fb(P)in (4), is proportional to the bistatic delay, and dependsupon five parameters (v1, rz1, tz1, v2, rz2). Eventually, werearrange the DPH as follows

fb ¼2pv1

l

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt� tz1Þ

2þr2z1

v21

sþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit2

v22

v21

þr2z2

v21

s !

¼2pv1

l

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt� tz1Þ

2þ t 2

1

qþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimt2 þ t 2

2

q� �ð5Þ

where we have introduced the three normalised parameters

t1 ¼rz1

v1

; t2 ¼rz2

v1

; m ¼v 2

2

v21

We notice that the general shapes of both the DPH andFTH depend upon four parameters ft1, t2, m, tz1g and ascale factor. In the constant offset case [11], m ¼ 1 andwe have three parameters whereas only one parameter isneeded in the monostatic case, where t1 ¼ t2; m ¼ 1;tz1 ¼ 0.

3 Small aperture

Although the non-stationary, bistatic DPH (5) is quite morecomplicated than a monostatic one, we can still approxi-mate it by quadratic expansion, at least for very smallazimuth apertures, to obtain a simplified focusing. Letus start from the DPH in (5), and evaluate first theinstantaneous frequency

ft ¼1

2p

@fb

@t¼

v1

l

t� tz1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt� tz1Þ

2þ t2

1

q þmtffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

mt2 þ t22

p0B@

1CA ð6Þ

and then the second derivative, the fm-rate

k ¼ f 0t ¼v1

l

t21ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ððt� tz1Þ2þ t2

1Þ3

q þmt 2

2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðmt2 þ t 2

2

pÞ3

0B@

1CA ð7Þ

Fig. 1 Generic bistatic geometry

IEE Proc.-Radar Sonar Navig., Vol. 153, No. 3, June 2006200

We finally assume the fm-rate to be constant, k(t0), wheret0 is the time when the target is at the centre of theantenna beam, to allow azimuth focusing by means of anymonostatic technique, like the Range Doppler. The non-stationary behaviour could then be handled by tuning thefm-rate with azimuth in block-like fashion (provided thatits variation is smooth).

The limit of this technique depends both on the time span ofthe observed DPH and the actual slope of its second deriva-tive, that is k0(t0). In order to appreciate visually this secondaspect, we have drawn in Fig. 2 the DPH and its derivativein an extreme case of a system that is both strongly bistaticand strongly non-stationary. We used (5) by assuming:tz1 ¼ 0.4, t1 ¼ 0.34, t2 ¼ 0.7 . t1, m ¼ 2. The figure hasdashed and dotted lines to show the two monostatic hyper-bolas (MHs), corresponding to the two square roots in (5),together with the resulting DPH, in continuous line,whereas its first derivative, f0

b(t), is plotted below. We wishthatf0

b(t) approximates that of a monostatic system; thereforewe need it to be nearly linear, and this assumption appearsreasonable only when t is far from the vertices of the twoMH where we have fast changes of slope. When t approacheseither tz1 (¼0 in our case) or tz2, the monostatic-based focus-ing fails, unless the bandwidth is reduced to a very smallvalue. In this case, as well as in the case of wide bandwidth,we need to develop a different methodology, and this is themotivation for this paper.

It should be noted that the example has been exaggeratedjust to provide a visual understanding of the problem: thevertices of the two MH have been made quite sharp byusing small values of t1, t2. In practical spaceborne or air-borne systems, these values are two orders of magnitudelarger than those used. Notwithstanding, the impact ofa non-stationary Doppler rate on the focused impulse

response could still be appreciable, particularly for highresolution SARs. In a general case, we expect that thevertex of the DPH is somewhat between the vertices ofthe two MH (just in the middle for the constant offset), itslocation has been derived in close form in the appendixand can be used, together with (7), to appreciate the validityof the monostatic approach in the worst case.

4 Reduction of general bistatic to monostaticsurveys

In the previous section, we have seen a simple approach tofocus bistatic surveys. However, the shape of the FTH isclose to singular in several cases. It is interesting, at leastfor comprehension, to transform a bistatic survey into amonostatic one in order to characterise their real differ-ences. On one side, the monostatic survey is azimuth invar-iant, whereas the bistatic surveys can well be azimuthvarying. We need an azimuth varying tool that transformsthe bistatic surveys into monostatic ones to characterisethis change. The approach that we will follow is simpleand discussed systematically in D’Aria et al. [11]. If theinput (bistatic) data set is made just by a single pulse, itimplies that we have scatterers on the ground plane onlyat the intersection between the ground plane and the iso-chronal ellipsoid that has foci at the positions that thesource and the receiver had when the pulse was emitted.Then, we revisit this ground distribution of scatterers witha monostatic survey. The result of this survey, namely a dis-tribution of arrivals in time and space, will correspond to thespace varying 2D transfer function between the bistatic andthe monostatic surveys. Its characteristics will elucidate thenon-stationarity of the bistatic survey and will allow us tounderstand the changes with respect to a monostatic one.The same analysis goes for the case of motion compen-sation: the data are acquired from a trajectory and trans-ferred to another. This will allow us to discuss the casewhen the synthetic aperture is wide with respect to the vari-ation of the fm-rate, and the DPH cannot be approximatedwith a second-order curve as previously suggested. In thiscase, we shall see that, given the digital elevation model(DEM), it is possible to derive a generalised smile [11].Another point that should be considered is that with thisway of operation, it will be easy to accommodate for theantenna pointing at the receiver and transmitter ends toevaluate any amplitude effects. Furthermore, in the caseof a strong scatterer, its correct amplitude will be recoveredcombining together all the arrivals; however, we refer toFomel [15] for studies on the correct distribution of ampli-tudes and illumination.

4.1 Need of a DEM

Before entering simple geometrical analyses, we would liketo show how it is that focusing bistatic surveys may bedependent on the information about the location of the scat-terers in the 3D space. This effect is clarified in the exampleshown in Fig. 3. Two different bistatic configurations areshown: in case (a), the two sensors move along the sametrack, with a proper displacement; in case (b), they followtwo parallel tracks. Two targets Pa and Pb are drawn; theyare seen at the same (bistatic) time when the midpointbetween source and receiver is at azimuth t ¼ 0, asthey belong to the same isochronal ellipsoid with thesefoci. In case (a) of aligned tracks, the two targets (andany target on a circle C centred on the track) will havethe same (delay, Doppler) coordinates in the data space;hence the same FTH would result for both targets. In this

a

b

¢

Fig. 2 FTH for a generalised bistatic configuration

a The bistatic phase is described by the double square root expressioncorresponding to the sum of two monostatic termsb Phase derivative

IEE Proc.-Radar Sonar Navig., Vol. 153, No. 3, June 2006 201

case, the data would span a 2D space, like for monostaticSAR, and a DEM characterising the position of the targetson circle C would be useless. On the contrary, in case (b),when the axis of the ellipsoid is no longer aligned withthe sensor track, the third dimension becomes relevant, aseach target along circle C generates a different FTH in the(t, t) domain. Then, the 3D spatial imbedding is importantand a DEM would be required to reconstruct both theshape and the position of each FTH, in order to performproper focusing. An incorrect DEM would lead to defocus-ing (in that, using an incorrect FTH, we would neglect topick up and combine some of the available Dopplercomponents and we would pick up and combine somecomponents containing only noise).

These considerations may appear finicky, but with anincreasingly wider relative bandwidth of the SAR data,like the unprecedented PAMIR SAR [16], the quality ofthe focusing has to be greater, reaching the limit when therelative bandwidth approaches 2.

4.2 Focusing using the pull technique

We will provide an example of focusing a bistatic survey,with no constraints at all on the antenna aperture, underthe following assumptions: (1) targets lay on the flatground; (2) orbits are straight and parallel to the ground(but not necessarily parallel to each other) and coplanar(same altitudes for both tracks). We assume that sensorsmay have different velocities. Although the case is fairlygeneral, it is not the most general we can conceive.However, we shall see in the next section that the procedureproposed here is flexible and can be extended to all the othercases.

Let us assume the geometry sketched in Fig. 4. The figureshows the two sensors, S1 and S2, their track and the track oftheir midpoint. We assume a reference system with thex-axis oriented like the midpoint along-track (azimuth),O(t), the z-axis perpendicular with respect to ground (hereassumed flat) and the x-axis perpendicular to the yz plane,as usual.

Let us apply the linear superposition to the input data (inthe data space), and let us suppose that we have a single

spike, at azimuth, range time (t ¼ 0, t ¼ tb). This spike cor-responds to a distribution of scatterers located on an ellip-soid that has foci in the position of the sensors at theacquisition time (t ¼ 0). Notice that the axis of symmetryof the ellipsoid is, in general, not parallel to the x-axis.We assume that locally it slants at an angle c, shown inthe figure. The equation of the ellipsoid can be written inthe local reference (x0, y0, z) that is rotated of c withrespect to the reference (x, y, z) (the x0-axis is also shownin the figure)

x02

a2e

þy02

b2e

þz2

b2e

¼ 1

ae ¼ctb

2ð8Þ

h ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2e � b2

e

q) be ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffictb

2

� �2

�h2

r

The sensor location at t – 0, which corresponds to the foci iswritten in the same reference as

S1fx0 ¼ h; y0 ¼ 0; z ¼ hsg; S2fx

0 ¼ �h; y0 ¼ 0; z ¼ hsg

We suppose that the scatterers lie on the ground plane.Therefore for each spike in the data space, a uniform distri-bution results in the model space of scatterers along anellipse as shown in the figure

x02

a2e

þy02

b2e

¼ 1 �h2s

b2e

x02

a2þy02

b2¼ 1 ð9Þ

a ¼ ae

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 �

h2s

b2e

s

b ¼ be

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 �

h2s

b2e

s¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffib2e � h2

s

qð10Þ

which corresponds to an impulse in the data. The focusing isthen carried out by means of the ‘pull’ technique, summingalong each FTP correspondent to each point of the ground

a

b

c

Fig. 3 Two targets, Pa and Pb in different locations are imagedby two bistatic systems

a Sensors aligned along the same orbitb Sensors flying on parallel orbitsc In the first case focusing is actually 2D (DEM independent) as thesame FTH results from both Pa and Pb. Not so in the second case

1

2

¢

Fig. 4 Geometry of the generalised bistatic acquisition; thesensor orbits are parallel to the ground

IEE Proc.-Radar Sonar Navig., Vol. 153, No. 3, June 2006202

plane. To check the correctness of this approach, we cannow model back the scatterer along an ellipse in the dataspace by means of a tomographic approach. For each ofthe scatterers in the model space, that compose theellipse, we backproject (spread, push) an FTH in the dataspace. The sum of all these FTH reproduce the spiky dataset, as shown in Fig. 5.

4.3 Generalised smile

The way we convert a generalised bistatic system into amonostatic one plus a proper pre-processing step followsclosely the derivation in D’Aria et al. [11]. The geometrynow is much different, in that the targets are located on aplane in 3D space (the flat earth spanned by azimuth, andground range) and no more in the azimuth, slant rangereference. This situation is far from the constant-offsetcase (coincident orbits, equal velocity). As in D’Ariaet al. [11], we map the distribution of scatterers onto amonostatic survey, corresponding to a spike in the initialbistatic survey. Then, focusing will correspond first to theconvolution of the initial data with the generalised smile,that is, the ellipse in the ground plane back as seen fromthe monostatic trajectory of choice, to be followed by theusual monostatic focusing. In the following section wewill also show how even motion compensation can be

seen under this perspective, leading to an intrinsicidentification of motion compensation and bistatic focusing.

As the ground plane imprint of any monostatic SAR is acircle, we decompose the elliptical-shaped reflector on theground as a superposition of many circular-shaped reflec-tors, each of them tangent to the ellipse. The idea isexplained in Fig. 6.

The geometry is detailed in Fig. 7: the target P on theground belongs both to the ellipsoid coming from thebistatic survey and to the circle coming from the mono-static survey with centre in M. We represent the ellipse(9), intersection of the initial ellipsoid (8) with the groundplane of the scatterers, with an envelope (superposition)of circles. We have to impose a common tangent betweenthe ellipse and the generic circle. For each target on theellipse P(x0 ¼ a cosf, y0 ¼ b sinf, z ¼ 0), the tangentangle is

dy0

dx0¼ �

b cosf

a sinf

and the equation of the normal to the tangent in P (the linePC in Fig. 7b) is then

h� b sinf ¼a sinf

b cosfðj� a cosfÞ

aj sinf� ða2 � b2Þ sinf cosf ¼ bh cosf

the centre of the circle, C, is the intersection with the lineh ¼ j tan c. The coordinates of C are thus

Cðx0 ¼ j0; y0 ¼ j0 tanc; z ¼ 0Þ

j0 ¼ða2 � b2Þ sinf cosf

a sinf� b cosf tancð11Þ

The monostatic delay is proportional to the radius of thesphere centred in M (see Fig. 7a)

Mðx0 ¼ j0; y0 ¼ j0 tanc; z ¼ hsÞ

and passing through P

Pðx0 ¼ a cosf; y0 ¼ b sinf; z ¼ 0Þ

The delay is thus

tm ¼2

cjP�M j

t2mc2

4¼ ða cosf� j0Þ

2þ ðb sinf� j0 tancÞ2 þ h2

s ð12Þ

Fig. 5 An elliptical distribution of targets on the flat ground inthe model space (see the perspective view), corresponds to asingle spike in the data space

This spike results from the superposition of all the generalised FTHsoriginated by each of the impulses that compose the ellipse. Themarked irregular shapes of the FTHs are due to the extremely non-stationary bistatic configuration

Fig. 6 Spike in the data-space is the result of the elliptic reflector on the left, intersection of the bistatic isochronal ellipsoid and the groundplane

This elliptic reflector can be modelled as a superposition of circles, each of them being a spike in a monostatic acquisition on the rightRadius of the sphere is proportional to the monostatic delay

IEE Proc.-Radar Sonar Navig., Vol. 153, No. 3, June 2006 203

and thus we obtain a distribution of spikes to be added to themonostatic survey in positions located along the x-axis withabscissa equal to the distance jC2O0j

x ¼ j0=cosc ) j0 ¼ x cosc ð13Þ

We can finally write the expression of the generalised smilein the azimuth, time domain (in the data space)

t2mc2

4¼ ðacosf� xcoscÞ2 þðbsinf� xsincÞ2 þh2

s ð14Þ

where the dummy variable f should be removed by exploit-ing (11) and (13)

j0 ¼ða2 �b2Þsinfcosf

asinf�bcosf tanc¼ xcosc ð15Þ

and the parameters describing the ellipse, a, b, should bederived from the acquisition geometry, namely the sensorsdistance, h (slow time variable, now) and the bistatictime, tb, according to (8) and (9)

be ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffictb

2

� �2

�h2

rð16Þ

a¼ctb

2

ffiffiffiffiffiffiffiffiffiffiffiffi1�

h2s

b2e

s

b¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffib2e �h2

s

qð17Þ

As an example, Fig. 8a plots different smiles correspondingto different values of the rotation, c. Notice that the par-ameters of the generalised smile require information aboutthe DEM.

4.4 Discussion: constant offset

It is interesting to look at these generalised smiles to try toidentify their message. It is evident that if the antennapattern is such that only a small part of the ellipse in theground plane is illuminated, these complex shapes maysimplify to the usual chunks of hyperbolas and thussimple second-order approximations of the FTH (the usualsmiles, then, maybe slightly tilted) could do in most ofthe cases. However, we also see that in some situations,namely correspondent to the cusps of the generalisedsmile, the second-order expansion would fail and we haveto move to more complex descriptions. In some cases, thebistatic configuration does not lead to a simple represen-tation as a monostatic one. We notice that these cuspsexist also in the case of constant offset, namely in theusual smile as discussed in D’Aria et al. [11]. However,in that case, the cusps correspond to energy backscatteredfrom points that are approximately along the survey line,that is, with extreme squint, and therefore not illuminatedwith the usual antennas-radiation patterns. The possibilityof opportunity bistatic surveys, where the geometry istotally random, may make these observations useful.

Another point that we just mention here has also beenwell discussed in geophysics [17]. This is the azimuth res-olution enhancement, made possible by combining multiplebistatic surveys. In fact, combining different transfer func-tions, one could increase resolution as with multiplesampling, thus gaining the equivalent advantage of anincreased PRF. This result has been independently rediscov-ered in the SAR community, pointing out the advantages ofa split receiver antenna for the reduction of the azimuthambiguities [3, 18].

Finally, a point that should be better analysed, is theimpact of the precision of the DEM on focusing. From(15), within a well-defined geometry, some conclusionsmight be drawn, which, however, are much too geometry-and resolution-dependent to be discussed in this paper,where we only would like to establish a general framework.

For better clarity, let us now verify that the smile deri-ved in D’Aria et al. [11] is a particular case of the general-ised smile derived here, for the constant-offset case. We

a b

1

2

¢

¢

¢

Fig. 7 Geometry for the reduction of the bistatic survey tomonostatic in the general case

a b

m m

Fig. 8 Plot of different generalised smiles, for different values of the ellipse rotation, c

a (t, x) data domainb (kx, t) data domain

IEE Proc.-Radar Sonar Navig., Vol. 153, No. 3, June 2006204

have parallel tracks, hence c ¼ 0, and (15) can be easilyinverted

x ¼ða2 � b2Þ cosf

a

cosf ¼ax

a2 � b2ð18Þ

that combined with (14), yields

t2mc2

4¼ a

ax

a2 � b2� x

� �2

þ b2 sin2 fþ h2s

¼ �b2x2

a2 � b2þ b2 þ h2

s

¼ �b2ex

2

a2e � b2

e

þ b2e ð19Þ

where we have expressed the parameters of the ellipseon the ground as a function of the ellipsoid parameters byexploiting (8) and (9). We furthermore notice thath2 ¼ ae

2 2 be2, and from (16)

t2m ¼ t2b �4h2

c2

� �1 �

x2

h2

� �

¼ t20 1 �x2

h2

� �

that is the smile, an ellipse in the (tm, x) domain. The smileis the plot for c ¼ 0 in Fig. 8a.

4.5 Generalised smile in k–t coordinates

Using the same technique that we have used in D’Aria et al.[11], we can now move the generalised smile into the wave-number time domain, simply by parameterising the smile interms of the time derivative

kx ¼dtm

dx¼

@tm@f

�@f

@x

rather than in terms of the abscissa x. The resulting functionis again double-valued and can be easily determined fromthe previous considerations. As an example, (18) and (19)have been exploited to compute the frequency domainexpression of the delays

tmðkxÞ ¼ tmðxðfÞÞ þ xðfÞ � kxðfÞ

represented in Fig. 8b.

5 Motion compensation as bistatic focusing

As observed in the previous sections, we can also cover inthe same way the case of motion compensation, that is,

the transfer of the data from one initial trajectory wherethey had been acquired to any other where they have tobe focused. The geometry is shown in Fig. 9. The largecircle on the ground underneath the sensor corresponds tothe uniform distribution of scatterers that contribute to theinitial survey with a single impulse at fast time t0. Itsradius is

a ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffict0

2

� �2

� z20

rð20Þ

where z0 is the sensor height. We assumed that only an arc,marked as a thick line in the figure, is effectively illuminatedby the antenna beam. The other two circles represent twouniform distributions of scatterers that contribute as twoimpulses in the second monostatic acquisition along thereference track. Their delays, to be compensated by thegeneralised smile, are proportional to the distances r1 andr2, in general: r(x). The monostatic delay is then

ctm

2¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiz2

1 þ pðxÞ2q

ð21Þ

where p(x) is the horizontal projection of r(x), and can becomputed based on the geometry of Fig. 10a. The figure

m

Fig. 11 Smiles, upper plot (marked as ‘þ ’), and frowns, lowerplots (marked as ‘W’) corresponding to the motion continuationcase, for different distances with respect to the reference track, d

a b

x

¢ ¢

d

Fig. 10 Motion compensation as bistatic focusing

a Derivation of the motion compensation kernel, to be compared withFig. 7b Delays from the portion of the target y . d decrease with jxj,generating a frown-like shape

1

2

Z

Z

y

x

0

d 1 2O

Fig. 9 Geometry for the derivation of the motion compensationoperator as a generalised smile

IEE Proc.-Radar Sonar Navig., Vol. 153, No. 3, June 2006 205

compares with Fig. 7b, now the single circular reflectorcentred on O is to be replaced by the envelope of circlescentred on the ground projection of the reference track, theline y ¼ d. The distance p(x) in (21) is either jP0 2 Cj orjP2 Cj depending on the sign of y, that is, the right/leftlooking of the sensor. We obtain

pðxÞ ¼ a+ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þ d2

pto be combined with (20) and (21) to derive the generalised‘smile’ plotted in Fig. 11. We notice that these kernels mayhave either the shape of a smile or a frown. This second casecomes from the arc corresponding to y . d as shown inFig. 10b.

Concluding this short section, we have shown that motioncompensation, that is, to use the geophysical jargon, thecontinuation of the data from one sensor’s trajectory toany other, can be carried out by convolving with a general-ised smile as shown in Fig. 11. Notice that, also, in the caseof motion compensation, the parameters of the generalisedsmile require information about the DEM.

6 Conclusion

In this paper, we have discussed the bistatic SAR surveyfocusing and the possibilities to transform a bistaticsurvey into the more usual monostatic one. We haveshown how to extend and generalise the concept of smile(the transfer function between bistatic and monostaticsurveys) to non-stationary acquisition geometries and tomotion compensation; the structure of these generalisedsmiles shows the existence of cusps that indicate that forgiven illumination patterns and directions of flight sucha reduction might be difficult. The cusps exist also in theconstant-offset case, but then they are limited to extremesquints and therefore not usually illuminated. However, inopportunity surveys, the geometry could be out of control,and therefore such situations may arise. We have alsoshown that a DEM is useful to focus on bistatic data (butonly if the offset is not constant and along the flight path)and in the case of motion compensation. The impact ofthe DEM imprecision, being too dependent on the bistaticgeometry and the resolution, is a priori unpredictable.

7 Acknowledgment

The authors would like to thank the anonymous reviewerswho helped to improve this paper with careful, completeand detailed revision.

8 References

1 Griffiths, H.D.: ‘From a different perspective: principles, practice andpotential of bistatic radar’. CIE-Int. Conf. Radar, 2003, pp. 1–7

2 Massonnet, D.: ‘The interferometric cartwheel, a constellation of lowcost receiving satellites to produce radar images that can be coherentlycombined’, Int. J. Remote Sens., 2001, 22, (12), pp. 2413–2430

3 Krieger, G., Gebert, N., and Moreira, A.: ‘Unambiguous SAR signalreconstruction from nonuniform displaced phase center sampling’,Geosci. Remote Sens. Lett., 2004, 1, (4), pp. 260–264

4 Yates, G., Horne, A.M., Blake, A.P., Middleton, R., and Andre, D.B.:‘Bistatic SAR image formation’. Eur. Conf. Synthetic Aperture Radar,Ulm, Germany, 25–27 May 2004

5 Cantalloube, H., Wendler, M., Giroux, V., Dubuois-Fernandez, P., andKrieger, G.: ‘Challanges in SAR processing for airborne bistaticacquisitons’. EUSAR’04, Ulm, Germany, 2004

6 Ender, J.H.G., Walterscheid, I., and Brenner, A.R.: ‘New aspects ofbistatic SAR: processing and experiments’. Int. Geoscience andRemote Sensing Symp., Anchorage, Alaska, 20–24 September 2004

7 Ender, J.H.G.: ‘A step to bistatic SAR processing’. Eur. Conf.Synthetic Aperture Radar, Ulm, Germany, 25–27 May 2004

8 Neo, L., Wong, F.H., and Cumming, I.G.: ‘Bistatic sar processingusing non-linear chirp scaling’. Proc. CEOS SAR CalibrationWorkshop, 2004

9 Sanz-Marcos, J., and Mallorquı, J.J.: ‘A bistatic SAR simulator andprocessor’. EUSAR’04, Ulm, Germany, 2004

10 Soumekh, M.: ‘Wide bandwidth continuous wave monostatic/bistaticsynthetic aperture radar imaging’. Int. Conf. Image Processing, 4–7October 1998, vol. 3, pp. 361–365

11 D’Aria, D., Guarnieri, A., and Rocca, F.: ‘Focusing bistatic syntheticaperture radar using dip move out’, IEEE Trans. Geosci. RemoteSens., 2004, 42, (7), pp. 1362–1376

12 Ding, Y., and Munson, D.C., Jr.: ‘A fast backprotection algorithm forbistatic SAR imaging’. Int. Conf. Image Processing, Rochester, 22–25October 2002, vol. 2, pp. 441–444

13 Loffeld, O., Nies, H., Peters, V., and Knedlik, S.: ‘Models and usefulrelations for bistatic SAR processing’. Int. Geoscience and RemoteSensing Symp., Toulouse, France, 21–25 July 2003, CDROM

14 Deregowski, S., and Rocca, F.: ‘Geometrical optics and wave theoryof constant offset sections in layered media’, Geophys. Prospec.,1981, 29, pp. 374–406

15 Fomel, S.: ‘Theory of differential offset continuation’, Geophysics,2003, 68, pp. 718–732

16 Ender, J.H.G., and Brenner, A.R.: ‘PAMIR – a wideband phased arraySAR/MTI system’, IEE Proc., Radar Sonar Navig., 2003, 150, (3),pp. 165–172

17 Bolondi, G., Loinger, E., and Rocca, F.: ‘Offset continuation ofseismic sections’, Geophys. Prospect., 1982, 30, pp. 813–828

18 Suss, M., Grafmuller, B., and Zahn, R.: ‘A novel, high resolution,wide swath SAR system’. Int. Geoscience and Remote SensingSymp., Sydney, Australia, 9–13 July 2001, pp. 1013–1015

9 Appendix: Zero Doppler bistatic delay distance

We compute the vertex of the DSR equation described by(5) in a very general case. The minimum of the DSR corre-sponds to the zero of ft, from (6)

t� tz1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt� tz1Þ

2þ t 2

1

q ¼ �1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

mt 2 þ t 22

p mt

ðt� tz1Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffimt2 þ t2

2

q¼ �mt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðt� tz1Þ

2þ t2

1

qð22Þ

We square both terms, obtaining a fourth-order equation.Care should be taken as solutions of the squared equationare not necessarily solutions of (22)

t4ðm2 � mÞ � 2tz1t3ðm2 � mÞ þ t2ððm2 � mÞt2

z1 � t22

þ m2t21Þ þ 2ttz1t

22 � t2

z1t22 ¼ 0

Notice that the solution degenerates for the constant offsetcase: m ¼ 1; however, we do not care for this case, asthere is already a closed form solution. We can make thereplacements

x ¼t

tz1

m ¼t2

2

ðm2 � mÞt2z1

k ¼ 1 �t2

2

ðm2 � mÞt2z1

þm2t2

1

ðm2 � mÞt2z1

and the expression further simplifies as a function of twoparameters

x4 � 2x3 þ kx2 þ 2mx� m ¼ 0

The solution is actually a function of two parameters. Wehave a fourth-order equation with four solutions, of whichtwo are real, and only one is consistent with (22). The

IEE Proc.-Radar Sonar Navig., Vol. 153, No. 3, June 2006206

solutions are

x1 ¼1

2�

1

6

ffiffiffi3

pg þ

1

6

ffiffiffi3

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6fg � 4kfg � gf 2 � gk2

þ12ffiffiffi3

pfmþ 6

ffiffiffi3

pfk � 6

ffiffiffi3

pf

fg

vuuut

x2 ¼1

2�

1

6

ffiffiffi3

pg �

1

6

ffiffiffi3

p

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6fg � 4kfg � gf 2 � gk2

þ12ffiffiffi3

pfmþ 6

ffiffiffi3

pfk � 6

ffiffiffi3

pf

fg

vuuut

where

p ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi81m2k2 þ 162m3k � 162m2k þ 3mk4 þ 81m4

�162m3 þ 3m2k3 þ 81m2 � 3mk3

s

f ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi54mk þ 54m2 � 54mþ k3 þ 6p3

p

g ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3f � 2kf þ f 2 þ k2

f

s

IEE Proc.-Radar Sonar Navig., Vol. 153, No. 3, June 2006 207