1362 IEEE TRANSACTIONS ON GEOSCIENCE AND … · 1362 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE...

1362 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 42, NO. 7, JULY 2004

Focusing Bistatic Synthetic ApertureRadar Using Dip Move OutDavide D’Aria, Andrea Monti Guarnieri, and Fabio Rocca

Abstract—The appearance of new synthetic aperture radar(SAR) acquisition techniques based on opportunity sources en-hances interest in bistatic geometries. In seismic data acquisition,each source is currently accompanied by up to 10 000 receivers,and in the last two decades, the bistatic geometry has been care-fully studied by scores of authors. Rather then introducing newfocusing techniques, within the first-order Born approximation(no multiple reflections), seismic bistatic acquisitions are trans-formed into monostatic ones using a simple operator named “dipmove out” (DMO). In essence, the elliptical locus of the reflectorscorresponding to a spike in the bistatic survey is forward modeledas if observed in a monostatic one. The outcome of the model, theso-called smile, is a short operator, slowly time varying but spacestationary. To transform a bistatic survey into a monostatic one,it is enough to convolve the initial dataset with this smile. Basedon the well-known similarity between seismic and SAR surveys,DMO is first described in its simple geometric understanding andis then used in the SAR case. The same processing that is beingused for movement compensation can be applied to the bistaticto monostatic survey transformation. Synthetic examples are alsoprovided.

Index Terms—Focusing, multistatic scattering, synthetic aper-ture radar (SAR).

I. INTRODUCTION

I N THIS PAPER, we shall try to pass to the radar commu-nity the understanding of bistatic surveys in terms of the

so-called “dip move out” (DMO), well known in seismics sincethe early 1980s; in doing that, we will use material published byseismic professionals in the 1980s and 1990s [1]–[4], but alsousing as long as possible, radar terminology besides the seismicone.

The renovated interest in bistatic radar surveys is due to sev-eral new acquisition techniques, where the illuminator can be anopportunity one. Typically, in the cartwheel [5] approach, threeto four “microreceivers” are orbited close to a satellite syntheticaperture radar (SAR) system, for instance trailing it at a distanceof a few tens of kilometers. Their orbit is not exactly identical, sothat they appear to describe an ellipse to an observer that tracksthe transmitter. For a sizeable part of the total time, a couple ofreceivers is in the proper position for interferometry, be it alongor across track. The processing to be done for the focusing ofthe data received by these microreceivers is close but not iden-tical to that correspondent to the zero offset (distance betweensource and receiver) acquisition, i.e., where the receiver is posi-tioned very close to the transmitter.

Manuscript received September 8, 2003; revised April 6, 2004.The authors are with the Dipartimento di Elettronica e Informazione, Politec-

nico di Milano, 20133 Milan, Italy (e-mail: [email protected]).Digital Object Identifier 10.1109/TGRS.2004.830166

There is never such a case of zero offset, since the wave prop-agates in a small but nonzero time, to return to the source afterit has moved of , i.e., a few meters, in the usual satellitesituations. So, if the minimum offset in the satellite case is say afew meters, the maximum envisaged in the case of the cartwheelcould be several tens of kilometers. Greater offsets are madeunlikely by the decreasing backscatter amplitude in directionsfurther apart from the specular one. Other situations correspondto the case of an orbiting illuminator, say a global positioningsystem transmitter, and a ground-based receiver. The case whenthe offset is time varying should be studied specifically. How-ever, we notice that the change in offset should be consideredduring the time that the target dwells within the footprint of thetransmitter’s antenna (or the receiver’s if it is narrower), and thistime might not be very large. Typically, the motion of the cart-wheel during a footprint is very small and will be neglected inour analysis. Thus, we will be able to study the situation as if theoffset was time stationary, even if this is not exactly the case.

So, we consider stationary situations, i.e., where the sourcedescribes a straight line, and the receiver follows the source ata given offset. The results come from the Born approximation,where multiple reflections are systematically neglected; in otherwords, we are supposing that each reflector acts as a point scat-terer, independently of the others. In seismics, where the av-erage reflection coefficient is a few percent (changes in rocksvelocities or densities are never that great, unless at the air– orsea–terrain interface), this is totally acceptable. Not so in theelectromagnetic case, where many strong scatterers exist, thatoften have a reflection coefficient close to one, and multiple re-flections within or without the same object are not negligibleat all. A splendid pictorial description of such a situation hasbeen recently made available [6] by the results of a very high(0.1 m) resolution radar system (PAMIR) where the physics ofthe scatterers is well understandable and the gigantic dynamicsof the electromagnetic reflectance in presence of conductors iswell visible; no such a thing exists in seismics, where the worldis transparent to the elastic waves that cross it. It should be wellunderstood that the entire concept of the possibility of trans-forming a bistatic survey into a monostatic one is meaningfulonly in the case of small reflection coefficients. Otherwise, onecould concoct counterexamples, where objects lose or gain vis-ibility depending on the bistatic angle of observation, enteringinto the stealth problematic, that we shall totally neglect here.

Apart from the seismic literature, the problem of focusingbistatic radar data has been studied by several authors; thefirst relevant papers on that problem appears to be due toSoumekh [7], [8] who approaches the problem in terms of thecross correlation with the hodograph of the data (i.e., theirDoppler history). Following Yilmaz and Claerbout [9], who

0196-2892/04$20.00 © 2004 IEEE

D’ARIA et al.: FOCUSING BISTATIC SYNTHETIC APERTURE RADAR USING DIP MOVE OUT 1363

first approached the problem in the open seismic literature, inthe sequel this hodograph will be referred to as the “doublesquare root equation” (DSR) from the two square roots thatappear in the expression of the travel path. These two squareroots reappear in other forms moving to the wavenumberdomain [1], [7]. However, the key advantage of the DMOtechnique, namely the possibility of decomposing the complex,time-varying operator resulting from the cross correlation withthe phase history of the DSR, into the cascade of the usualmonostatic processing plus the application of a time-varying,but short operator (i.e., the DMO operator) is not mentionedin Soumekh [7], even if the DMO operator shows up in a lineof the derivation (19). In other papers, the same author haspublished more considerations on the DSR, be it in the spaceor in the wavenumber domain. Munson [10] has discussedthe problem in terms of the backprojection, that is the adjointoperator. Again following Claerbout [11], we can refer topull or push operators depending on whether we consider animpulse in the model domain (then, to recover the model, wecorrelate the image in the data domain with the DSR) or animpulse in the data domain and then we spread (backproject)its amplitude in the model domain. The cascade of the twooperators should be a delta function; if it is not, we shoulduse an additional operator (the so-called rho-filter) to equalizeamplitudes, outside the null space. The book of Tarantola [12]is a very good reference for these considerations. More authors[13]–[15] are now studying the problem within the radarcontext, and therefore we think it could be useful to reanalyze,using also the radar geometry, the problem of constant offsetacquisitions.

After a short discussion of the complete push and pull oper-ators (the ellipse and the DSR), we will introduce the conceptof DMO in Sections IV and V, first in the space domain andthen in the wavenumber domain. The analysis of its behaviorwith radar frequencies and geometries will be discussed in Sec-tion VIII, and an example of the results of the processing will begiven in Section XII, together with a simple recipe to transforma code for Motion Compensation using subapertures into a codefor bistatic focusing. Both types of codes share the time-varyingcharacter as well as the nature of being a prequel to a monostaticfocusing procedure, be it anyone of choice.

II. FOCUSING MONOSTATIC SURVEYS

To get familiar with the way of thinking of the geophysicists,we first consider the well-known monostatic surveys. We con-sider first the data space, correspondent to the domain where thedata are acquired. As a function of the abscissa of the source,coincident with that of the receiver, we measure reflections at(monostatic) times . We distinguish the data (or signal)space , from the model space where the scatterers are,their position being a function of the abscissa again and of theslant range , distance from the trajectory of the source. If wehave one spike in the data space, at time , when the illu-minator and receiver is at the abscissa 0, then the scatterers arelocated along the circle

(1)

Fig. 1. Focusing, as seen in the data space. A spike in the data correspondsto a circle in the model. Then, to each scatterer of the many that compose thecircle, a hyperbola corresponds to the dataset. The sum of all these hyperbolasrecreates the observed spike.

Similarly, if we have only one scatterer (a spike in the modelspace, now), then in the data space we measure reflections atthe times corresponding to a hyperbola

(2)

To understand the focusing technique we start with a singlespike in the data space. In the case that the velocity of themedium is the constant , then we know that the only possiblepositions of the reflectors in the model space are along the circle(1). If we had only one scatterer with coordinatesthen the echoes would be located in the data space along thehyperbola in the coordinates (2). So, if we have asingle scatterer, we measure one hyperbola; if we superpose theeffects due to the many scatterers that make the circle, we haveto superpose the correspondent hyperbolas in the data spaceand we get back the initial spike, as shown in Fig. 1.

Vice versa, suppose we start with a single scatterer in themodel space; we see a hyperbola (only one, now) in the modelspace, and then we expand each measurement into the corre-sponding circle in the model space. Again, all the circles willsuperpose into the initial position of the scatterer (Fig. 2).

In conclusion, to focus the data, following the tomographic(push) approach, we spread the incoming pulses of the dataspace along circles (1) in the model space; alternatively (the pullapproach), we collect all the echoes relative to a given point scat-terer by averaging the amplitudes in the data space along the cor-respondent hyperbolas (2) (focusing by cross correlation). Thisis shown in Figs. 1 and 2, which demonstrate the reconstruc-tion of the pulse. These considerations are purely kinematical;it is also well known that, unless the proper weighting along cir-cles or hyperbolas is adopted, there is a residual defocusing thatshould be dealt with the so-called rho filtering [11]. Besides, thespectral coverage is dependent on the dips of the hyperbolas orcircles that contribute to the formation of the spike [11].

The key point is that the cross correlation with the time-varying hyperbola operation that could appear computationally


Fig. 2. Focusing, as seen in the model space. A point scatterer generates ahyperbola in the monostatic dataset. The circles that correspond to the spikesthat compose the measured hyperbola superpose on the initial point scatterer.

Fig. 3. Geometry of the bistatic system.

heavy is made cheap and simple by well-known focusing tech-niques like the Stolt interpolation [16]–[18].

III. FOCUSING BISTATIC SURVEYS

To focus bistatic surveys, we will use the same procedure asfor the monostatic approach, supposing the reflectors to be in-finitesimally weak, and then using the superposition principle.So, now let us consider a bistatic survey (the geometry is indi-cated in Fig. 3), and let us indicate with , the positions of thesource and receiver.

We suppose that the receiver trails the source at a constantoffset

(3)

The midpoint between source and receiver is indicated againwith

(4)

So, if , we are back to the monostatic case. Any bistaticdataset , where is the azimuth and the bistatic ar-

rival time, can be decomposed, using the superposition prin-ciple, in a superposition of pulses. Without loss of generality,as before, we suppose that we measure only a reflected pulse, atthe time and when , i.e., at the position .We suppose again that the velocity of the medium is a constant. Then, we can say that the reflectors that created this echo are

spread along an ellipse in the model space. This is the two-di-mensional (2-D) physical space that has for horizontal coordi-nate the same as for the data space, but which has as verticalcoordinate the spatial coordinate (slant range) and not the timeof the arrival as in the data space. This ellipse has foci in the po-sitions and ; the length of its horizontal semiaxis is

(5)

and that of the vertical semiaxis is

(6)

In fact, the total travel time from any point of this ellipse tosource and receiver adds up to . Any other point in the modelspace corresponds to a different value of total travel time. Sincewe are supposing that the elements of the ellipse are point scat-terers, it is evident that this analysis is purely kinematical andnot dynamical; in other words, we are not attempting to appre-ciate amplitudes. Thus, we are not trying to calculate the reflec-tivity of the elements of the ellipse, that needs not to be constant.Besides, the thickness of the ellipse is also not considered andthe consequent slant range resolution; however, this is typical ofthe stationary phase approach, and correspondingly of the op-tical ray theory.

We note in passing that a lot of attention has been dedicated,in the seismic literature, to the amplitudes problem [4], [19]; webelieve that this problem is still premature for radar.

As observed, we indicated with the horizontal axis (thelongest, along azimuth) of the ellipse and with its shortest axis,vertical, along-slant range. Using the current coordinates and

in the model space, the equation of the ellipse is

(7)

As the ellipse is the correspondent of the circle for , incorrespondence of a single point scatterer in the model space,we have a flat-top hyperbola in the data space instead of thehyperbola seen before. In fact, if we have a single scatterer atabscissa 0 and at slant range in the model space, in the dataspace we will see reflections at the times given by the so-calledDSR equation [11] (see Fig. 5)

(8)

The focusing comes from the superposition of effects; inFig. 4, we see the effects of a spike in the data space. There isan ellipse in the model space, and each of its points correspondsto a different flat-top hyperbola, again in the data space. Theirsuperposition confirms that there was just the initial spike inthe data space. In Fig. 5, we see the effects of a spike in themodel space. Thus, there is only one flat-top hyperbola in themodel space, and each of its points corresponds to a different


Fig. 4. Spike in the bistatic dataset corresponds to an ellipse in the model.Then, to each scatterer of the many that compose the ellipse, a flat-top hyperbolacorresponds to the dataset. The sum of all these flat-top hyperbolas recreates theobserved spike.

Fig. 5. Point scatterer generates a flat-top hyperbola in the bistatic dataset.The ellipses that correspond to the spikes that compose the measured flat-tophyperbola superpose on the initial point scatterer.

ellipse, in the model space. Their superposition recreates theinitial spike in the model space. In both cases, we see that theflat-top hyperbola is the inverse operator to the ellipse, as thehyperbola was the inverse to the circle. This is good news andbad news.

1) Good news: Because we know now how to focus bistaticsurveys. It is enough (pull) to sum along flat-top hy-perbolas in the data space (correlation in data space) or(push) to spread each sample in the model space alongits corresponding ellipse (backprojection).

2) Bad news: Because these techniques are expensive andwe cannot reuse the focusing machinery developed formonostatic surveys (spreading along circles or summingalong hyperbolas).

IV. SMILE

Let us assume the same semielliptical distribution of reflec-tors in the model space that causes the spike in the data space

Fig. 6. Semielliptic reflector sketched as marked dotted line, whichcorresponds to a spike in a bistatic dataset, is observed by a set of monostaticsurveys (circles). Dots indicate the locations of the centers of the circles,corresponding to positions of arrivals of reflections from the elliptic target.

Fig. 7. Two-dimensional bistatic geometry, in the model space. The ellipticalreflector is approximated by an equivalent monostatic circular reflector. Thecenter and the radius of the monostatic observation, change as a function of thesquint angle.

shown in Fig. 4. In order to reuse the monostatic techniques,we need to express the same bistatic dataset as the superposi-tion of many monostatic datasets, (i.e., many circles centeredin different position along azimuth, and with different radii ordelays), as shown in Fig. 6. We may think as a semiellipticalreflector imaged by different monostatic surveys. Each of thesesurvey will show arrivals (specular reflections on the points ofthe semiellipse) in an interval of positions of thesource (coinciding with the receiver). If the source is outsidethis interval, there is no specular reflection. If the abscissa ofthe source is inside this interval, the reflections will arrive at thetimes when the expanding circular wave centered on the sourcelocation at the abscissa (within the interval) is tangent to theellipse and thus is mirrored back to the source. Let us refer tothe 2-D geometry sketched in Fig. 7. To find the arrival times ofthe specular reflections on the ellipse, we indicate with andthe coordinates either of the points of the circles correspondingto the monostatic survey or of the ellipse; indicating with themonostatic travel times and, thus, with the radii ofthe circles, we have

(9)


Fig. 8. Geometry for deriving the smile. The upper plot allows the calculationof the proper monostatic delay t for each azimuth y in the model space. Thesedelays are then applied in the data space (y; t ), as shown in the lower figure.The locus of the delays is the smile operator.

The equation of the location of the reflectors in the model space,i.e., the ellipse, is again

(10)

Imposing a common tangent to the circles and ellipse to repre-sent the specular reflection, we have

(11)

We equate the dummy variables and (the coordinates ofthe backscattering point on the circle and the ellipse) and theirderivative (the tangent of the angle of the elementary mirror)

(12)

(13)

Then, substituting back for the dummy variables and , we getthe monostatic arrival times to and from the ellipse as afunction of the position of the illuminator. If we now plot thearrival times as a function of the position of the illuminator,we get a line in the data space as Fig. 8 shows, which pertainsto an ellipse (yes, but another one, not that in the model spacewe started with) with equation (in the data space!)

(14)

This ellipse bottoms at time

(15)

for and touches the axis at the abscissas . Not all theellipse pertains to the locus, since for we haveand ; the locus covers only a small part of the ellipse(and has thus been named “smile” for its peculiar shape). Thehorizontal extension of the locus is the interval

(16)

Fig. 9. Smile is obtained from the envelope of the hyperbolas that correspond,in the data space, to each point scatterer that composes the ellipse, in the modelspace.

Within this extension, and since the offset is in general muchsmaller than the maximum monostatic slant range , it is gener-ally possible to expand the square root and approximate

(17)

Notice that the smiles (Fig. 9) due to a spike arriving at timebottom at time

(18)

In other words, a monostatic survey would see any scatterer atearlier times.

In conclusion, we have decomposed an ellipse in the modelspace into the superposition (the envelope, that is) of many cir-cles, again in the model space.

Then, instead of focusing the ellipse with a bistatic expensivetechnique, using a cheap monostatic technique we can focus thesuperposition of circles, provided that we replace each sample ofthe bistatic survey with the correspondent smile to transform thebistatic dataset into a monostatic one. Thus, with the convolu-tion with the smile, we have transformed the flat-top hyperbolainto the usual hyperbola (Fig. 10). With respect to the complex-ities of using the fully extended flat-top hyperbola to focus, wesee that there is the advantage that the smile is a short operator.We still have the disadvantage that the smile is time varying. Wewill see that this drawback can be overcome with a logarithmicstretch [2], [20].

V. WAVENUMBER DOMAIN FOCUSING: SLOPES

OF THE TANGENTS TO THE SMILE

Let us now see how to convolve with the smile, and in par-ticular, how to take into account the dip of the reflector or thesquint of the beam. In this section, we indicate with thebistatic travel time correspondent to the illumination of a re-flector slanting of the angle ; in radar terminology, is thesquint angle of a monostatic survey. If we have only horizontal


Fig. 10. Single scatterer in the model space generates a flat-top hyperbolain the bistatic dataset. If, instead of each point of the flat-top hyperbola theproper smile is positioned in the data space, their envelope will be the verysame hyperbola that corresponds to the initial point scatterer, but in a monostaticsurvey.

reflectors , i.e., for zero squint, to transform a bistaticsurvey that sees an arrival at time into a monostaticone, it is enough to anticipate the data in the data space of thetime

(19)

Suppose now we had a reflector dipping by the angle(hence, not a point scatterer). Then, the same results come ifthe beam is squinted fore or aft of the same angle . In thiscase of nonzero , the offset entails a smaller delay (DMO,in seismic terminology) as seen in Fig. 11. We indicate with

the monostatic travel time

(20)

If we consider the triangle XRS in the figure, we get

which leads to the following relation:

(21)

We then combine expressions (20) and (21)

(22)

This expression leads to the same smile operator, just derived inthe previous section

(23)

Fig. 11. Bistatic system geometry for computing the DMO operator. Thegeometry refers to a reflector dipping by the angle �, hence not a point scatterer.Notice the reflector point dispersal (see [21]).

Fig. 12. Smile decomposed in the envelope of its tangents. Instead of crossingin a point, these lines are delayed (moved out) differently depending on theirDMO. The external tangents pass through the dot that indicates the bistatic traveltime. In fact, 90 squint entails no delay change between the monostatic and thebistatic cases.

Notice that the total delay of a bistatic survey with respect tothat of a monostatic survey is maximal for zero squint

(24)

and goes to zero ( ) when thesquint is . Thus, the smile can also be seen as a negativedelay as a function of the squint of the beam , i.e., the Dopplerfrequency in radar terminology (Fig. 12). The same results canbe got directly from the smile equation, with some algebra. Infact, it can be shown that the tangent to the smile that has slope

(25)

intersects the line for

(26)


Therefore, instead of convolving with the smile, we can applydifferent negative delays as a function of the squint . This neg-ative delay is

(27)

(28)

Equation (28) agrees with (14): however, in one case (14), thedelay is seen as a function of the position of the source in themonostatic survey; in the other case (28) the delay is a functionof the squint of the beam. Thus, the smile is seen as the envelopeof its tangents, each of them corresponding to a different squint,i.e., a different Doppler frequency, i.e., a reflector with dip (ina short time window, to avoid the effects of the time variance).To convolve with the smile, we can then decompose the 2-Dinput dataset into several one-dimensional subaper-tures in the Doppler domain, apply to each subset a time-varyingnegative delay, and then recombine. This is easily done in thehorizontal wavenumber and time domain, or even better in thewavenumber-frequency domain, if we learn how to deal with thetime variance of the smile.

Let us express with equations the concepts that we have seenin this section, up to now. Let us indicate with the mono-static azimuth wavenumber, and express its relation with thesquint angle , slope of the reflector. The monostatic Dopplerfrequency is [see also (26)]

(29)

The squint angle is thus related to the Doppler frequency do-main variables

(30)

(31)

In the case of the smile, the dip as a function of the horizontalcoordinate is

(32)

The delay as a function of the Doppler frequency is then

(33)

(34)

Finally, solely by applying the negative delay as a function ofthe Doppler frequency

(35)

we transform the bistatic dataset into a monostatic one.

VI. LOGARITHMIC STRETCH

We still have to cope with the nonstationary behavior of thesmile with time. The easy way is to apply the so-called loga-rithmic stretch to the input data. We start from (28) to see thatif we pose

(36)

then

(37)

So, instead of a time-varying delay

(38)

we will apply the one

(39)

For , we can take the minimum two-way travel time of thesurvey. Indicating with the domain conjugate to that of thevariable , we can now operate in the , domain by applyingthe operator

(40)

which is nilpotent for , as it should [2], [20].Clearly, (40) is an approximation, and it can be shown that

the exact log-stretch phase shift is

where

(41)

Equation (41) was derived in different ways by Liner [22],Zhou et al. [23], and Canning and Gardner [24]. In [25], it isderived by solving the offset continuation partial differentialequation. The equation solution suggests an amplitude factor inaddition to the phase shift but it can be neglected if amplitudesare not important for data processing. The implementation ofthe log-stretch operator requires that data are resampled, inthe fast-time direction, according to (37); however, we assumeirrelevant the cost of this monodimensional operator in thebudget of the whole 2-D bistatic focusing.

VII. OFFSET CONTINUATION: APARTIAL DIFFERENTIAL EQUATION

It can be of interest to observe that there is another techniqueto focus bistatic data, named “offset continuation.” A partialdifferential equation can be found that links the datameasured at different offsets, but related to the same model inthe model space [2], [25].

(42)


The advantage of this approach comes when different re-ceivers have to be combined together into a single image. Then,the above equation acts as a train going toward the final stationat , leaving from . New data are loaded on thetrain (stacked) whenever the offset is the corresponding one. Inother words, let us start the integration of the equation using asinitial condition the data measured at . Then, we usethe equation to continue the data to a smaller value of , ,stack in the data from the second receiver that has , andso on, until the monostatic data are also stacked in, when finally

. If the receivers are in the thousands, this technique comesin handy.

VIII. RADAR GEOMETRIES

In the previous sections, we have decomposed an entire el-lipse into the superposition (the envelope, that is) of numerouscircles, transforming a bistatic survey into a monostatic one. Inthe radar case, and even more if the radar is spaceborne, the partof the reflectors ellipse in the model space that is illuminated bythe source is limited, since the beamwidth of the transmitting an-tenna is generally small. This is done in order to keep the SNRhigh enough and then to prefilter the data to remove the spatialalias and limit the spatial sampling rate to that correspondent tothe image resolution.

We should consider the fact that, if the Doppler spectrum islimited by the antenna, the same should happen to the azimuthalspectrum of the smile. It is not efficient to convolve two datasets,unless the spectra have the same support, especially if there arerisks of alias. Now, let us indicate with the central Dopplerfrequency. This need not to be zero, especially if the monostaticSAR is pointed broadside, toward zero Doppler, that is. Themonostatic survey correspondent to the bistatic survey, due tothe offset of the receiver, is pointed at Doppler

(43)

The same happens for the bistatic survey.It is important to observe that, as long as the rough approxi-

mation (28) holds, in this transformation from bistatic to mono-static surveys, there are no change of the Doppler frequency,but only phase shifts. Thus, the interferometric characteristicsof the data are not modified.

If the Doppler band of the bistatic survey is limited by thefootprint of the illuminator, indicating with the resolutionalong azimuth

(44)

The maximum and minimum squint angles , are

(45)

and consequently we can determine the minimum and max-imum azimuth wavenumbers. Depending on , we might verywell have that the bistatic survey has a spectrum totally disjointfrom that of the monostatic survey.

From this, comes the rationale of the cartwheel, since in thatcase the relative offsets are kept systematically small enough to

ensure the spectral overlap, at least between the data recoveredby the elements of the cartwheel. To move to the monostaticsurvey, we have to apply Doppler and time-dependent negativedelays (the smile). The maximum delay within the spectral sup-port of the data, limited by the onboard antenna is

(46)

(47)

(48)

(49)

(50)

In a typical case (L-band, 4-m azimuth resolution), we get thereassuring figure

(51)

In the extreme case of 0.1-m resolution and X-band, we have

(52)

and therefore 150 resolution cells. This number grows further ifthe main beam is squinted away from the bistatic receiver. Thisproves that with the expected values, DMO will not be a verytaxing radar processing.

IX. NONSTATIONARY SURVEYS

Up to now, we supposed that the offset between source andreceiver was constant; what if it changes during the survey? Ob-viously, we need to consider only the changes of the positionof the receiver while the same target stays within the footprintof the source antenna, so that some approximation can be ac-cepted. We consider first the case of a stationary illuminator. Inthis case, studied by the geophysicists as the Shot DMO [26], wehave that a spike measured at any given location and time corre-sponds to scatterers located along an ellipse that depends on thepositions of source and receiver as seen previously. In this case,the nonstationarity of the pulse response, neither in short norin long times, excludes the possibility of a convolutional solu-tion. The data consist of delayed one way hyperbolas, each onecorrespondent to a single scatterer. The solution proposed by thegeophysicists is tailored to be cast back into monostatic surveys,to simplify the data handling. Again, the general use of the Bornapproximation is still useful. Let us consider each pulse emittedby the source positioned in the location (bold, since it isa vector, as it is any position say on a plane, and a function ofthe “long” acquisition time ). It is backscattered by the re-ceiver positioned in the location and arrives at the time

. The locus of the scatterers is a rotational ellipsoid withfoci in , , and main axis with length . The application ofthe same DMO principle shows that a monostatic survey on thisdataset corresponds to a smile (a line in the three-dimensional


(3-D) space spanned by , , ) laying in the vertical plane con-taining , that bottoms in

(53)

More pulses in other positions , will correspond tomore smiles in the correspondent planes. The positions of theequivalent monostatic sources are not along a line any-more, but on a surface.

(54)

This entails the existence of nonzero baselines and thereforeinterferometric effects. Combining the dip moveout techniquewith that of motion compensation, it should still be possible totransform this dataset into that acquired by a monostatic survey,apart from the unavoidable interferometric effects. However, theproblem becomes longer to describe, and we will postpone itsanalysis to another paper. The problem of 3-D DMO has indeedlong been studied in seismics and can be extended to this case,when useful [4] .

X. STACKING SMILES

A further advantage of multistatic surveys comes from thepossibility of combining the data to improve the overall spa-tial resolution, limiting at the same time the overall data rate toefficient values. It is well known that a time series sampled atinterval can be subsampled times to obtain differenttime subseries that can be recombined to reproduce the originalone. Necessary condition for that is the “linear independence” ofthe subseries. In other words, all parts of the original spectrumshould be recoverable and the alias should be removable. Thesame concept applies to multistatic surveys: it is reasonable toexpect that say using three receivers instead of one, it should bepossible to have each of them using a PRF reduced at best to 1/3to recombine the data to recover the original. This may happensay if the midpoints of the three surveys are staggered in space,or if the effect of filtering due to the smiles is properly used [27],[28]. Obviously, to avoid spectral holes, the offsets should beproperly positioned. However, it could be an additional advan-tage of the cartwheel geometry, as it was observed from the be-ginning too [5], supposing that each platform observed a disjointpart of the entire Doppler spectrum. It is interesting, in the caseof DMO, to remark the effects of a point scatterer in the modelspace. The superposition of several coregistered bistatic surveystransformed into the monostatic one through DMO correspondsto a monostatic dataset where several smiles are stacked to-gether. Their envelope results to be part of the hyperbola thatcorresponds to the initial point scatterer [29].

XI. EVALUATION OF THE AMPLITUDES

Up to now, we have solved for the kinematical part of theproblem; it has not been clarified, however, which amplitudes(and phases) should be smeared along the smile. This impliessome hypotheses on the scattering behavior of the material in-vestigated. The problem is very important in geophysics, since

from the amplitude of the reflections, very valuable informationon the structure of the reservoir can be obtained, say determiningwhether there is a oil gas contact or not. The situation is not thatmature for radar data, where also the much greater dynamic ofthe data would make such an analysis much more difficult. Thebest known reference for that is the paper by Black [19] (alsoavailable in [4]).

XII. IMPLEMENTATION FOR MICROWAVE SAR

Let us come to the case of microwave SAR, where data aredown converted, time domain windowed, and sampled. If wedefine the time of the down converted data and its corre-spondent wavenumber, we need to replace in place of

in the implementation of the smile operator. This notation isconsistent with the Appendix: the operator in (35) and (85) isto be applied to each single contribute in the range compresseddata to convert the signal into the one achieved througha monostatic SAR. The resulting the time-varying shift1 is thus(85)

(55)

where

(56)

Eventually, we decompose the operator (55) into two terms

(57)

(58)

The first operator (57) is -varying, hence implements a mi-gration (in the received field): it corresponds to the azimuth-wavenumber-dependent delay of the smile (50). The second op-erator (58) is still the smile, but as a pure phase term, due to datademodulation. This operator is not present in the geophysicalformulation. However, the small fractional bandwidth of usualSAR makes the smile-phase (58) the dominant term. Notice thedependence on of this operator: in the case of a squinted SAR,the smile would provide both a range and an azimuth shift; thisazimuth shift is equal to the offset of the monostatic center inFig. 11.

The major problem in the implementation of (57) and (58) isdue to their time-varying nature.

The migration operator (57) is slow time-varying and canbe implemented blockwise (the smile is actually a very shortoperator), as a shift in the range-Doppler domain (or a linear

1According to the notation in the Appendix, we use to define the domainconjugate to the time of the converted monostatic field, t . This time differsfrom the bistatic time of the raw data t , due to the stretch implemented by thesmile. However, this stretch is so small in the duration of the echo that can beneglected; hence, we will use and ! with the same meaning.


phase in the , domain). As an option, the operator couldbe made time-invariant by exploiting the logarithmic stretch inSection VI. In most cases, it is sufficient to implement the delayat the nominal squint angle,

(59)

The positive phase slope remind us that the operator actually an-ticipates data depending on the wavenumber, as already shownin Fig. 12.

The phase operator (58) is monodimensional and can be up-dated at each range bin in the range Doppler domain. How-ever, it is safe to keep the full expression, with no approxima-tions

(60)

The limitation embedded in (60) is due to the variation of thesmile delay within one range resolution cell, which should givea negligible phase error. If we approximate the operator (58)

(61)

and we impose that the phase error to be within in one reso-lution cell, ( being the range bandwidth), we get

(62)

As an example, in a microwave SAR with 20-MHz bandwidthand operating in C-band (5.3 GHz), the ratio between the trans-mitter–receiver separation and the bistatic closest approachshould be lower than 0.06, corresponding to a closest approachlarger than 300 km with a bistatic distance km. Noticethat the limit becomes less stringent as the bandwidth increases.

Finally, notice that as the phase operator (58) is time-varying.It cannot be simply interchanged with the premigration (57): ifwe implement first this second one, then we should account forit in the parameter when implementing (58).

A. Focusing

The bistatic raw data, converted into an equivalent monostaticfield by the preprocessing just described, can be focused by anystandard technique already developed for monostatic SAR, butsome tuning of the processor parameters is required.

First, the spectral support in azimuth should be the one thatcomes out from the bistatic acquisition (we stress the fact thatthe smile operator does not perform any wavenumber conver-sion); therefore, the central wavenumber, or the Doppler cen-troid, should be computed from the bistatic geometry of the ac-quisition system. We need to derive with respect of azimuth timethe Doppler phase history described by the flat-top hyperbola

(63)

TABLE I

where is the monostatic closest approach (e.g., with respectto the center of the bistatic system), and corresponds to theazimuth, , ( being the slow time), as shown in the geometryof Fig. 3. The bistatic Doppler rate is

(64)

where the angles and are defined in Fig. 11. The actualDoppler centroid should be computed by (64) in correspondenceof the azimuth time that marks the center of the beamwidth inthe acquisition geometry (the illuminated portion of the flat-tophyperbola).

As a second aspect, we must remark that, although the smileperforms a (nonuniform) compression of the received image,the time reference of the output is not to be changed. This im-plies that the focusing processor should compute the locationof the focused targets by using the bistatic time labeling, andnot the one that would come out from the monostatic-converteddataset. In other words, for an unsquinted geometry, a hyper-bola whose vertex is at range bin corresponds to a target withclosest approach

(65)

( being the sampling window start time), and notas one would expect in a monostatic

system. In most cases, no change is needed in the processor,but a simple constant shift of the output time reference. Incase of large swaths and/or wide bistatic apertures , this shiftshould be made range variant, which would be quite easyaccomplished in a range-Doppler processor, whereas it wouldrequire a modification of the Stolt interpolation or of the chirpscaling function in a wavenumber domain processor (see [17]and [18]).

B. Results From Simulations

A preliminary validation of the technique here presented hasbeen made by numerical simulation. A C-band bistatic SAR hasbeen designed by assuming the usual geometry in Fig. 3, andwith the parameters in Table I. The choice of parameters hasbeen made in order to check a worst case condition, where rangemigration is considerable and the bistatic system is quite dif-ferent from a monostatic one. The simulated dataset has beenpreprocessed as described before and focused by a standard


Fig. 13. Focusing a set of point targets using the proposed preprocessing.Amplitude of the focused image in decibels. The targets are well focused and inthe correct location (marked as a star in the image). Squint angle � = 0.

Fig. 14. Focusing a set of point targets without using the proposedpreprocessing and just tuning the monostatic closest approach, same dataset asthe one assumed in Fig. 13. The targets are defocused and not properly located.

wavenumber-domain monostatic processor (with a careful Stoltinterpolation).

The result achieved after focusing a set of point targetsaligned along azimuth at squint angle (hence, ),and equally displaced in slant range, is shown in Fig. 13. Thetargets are well focused, getting the resolution compatible with

Fig. 15. Focusing a set of point targets by exploiting the proposedpreprocessing. The targets are well focused and in the correct location (markedas a star in the image). Squint angle � = 10 .

Fig. 16. Focusing a set of point targets without using the proposedpreprocessing, same dataset as the one assumed in Fig. 15. The targets aredefocused and not properly located.

the spectral support and almost in the correct position, therange displacement due to the approximation implied in (65)being hardly noticeable. As a comparison, the result achieved


by processing the dataset as monostatic, e.g., by assuming thehyperbola that “approaches” the flat-top one, is in Fig. 14.

As a further examples, the same bistatic system has been as-sumed, but with a squint angle . The result is reported inFigs. 15 and 16: note that without the proper processing, targetsare defocused (less than for zero squint as the bistatic effect isless marked), and also misplaced both in range and in azimuth.

XIII. CONCLUSION

In the paper, we have shown that the framework developed ingeophysics since the early 1980s for modeling and processingbistatic (and multistatic) systems can be ported with success tothe case of microwave SARs. This framework establishes a re-lation between a bistatic system and an equivalent monostaticone centered in the midpoint of the two sensors. The conver-sion is performed with no approximation by a time-varying pre-processor, which acts as a sort of motion compensation (actu-ally it can be used also for motion compensation). Preliminarysimulations confirms the feasibility of the described techniqueeven in worst case conditions, of squinted systems with largemigration and significant bistatic apertures. Hints are proposedto extend the idea to nonstationary systems (not the case of fu-ture spaceborne constellations).

We think that the principal advantage introduced by the pro-posed bistatic-to-monostatic conversion is not the reuse of theexisting the SAR processors, but rather the extension to thebistatic SAR of the zero-Doppler coordinate systems, with theensuing simplifications in the design of future spaceborne sys-tems and in the geolocation.

APPENDIX

DMO DERIVATION BY STATIONARY PHASE

In this Appendix, we derive the decomposition of the forwardbistatic model into the equivalent monostatic one and the DMOoperator by a formal point of view. The derivation here shownis a porting to the SAR community of the work of [30], whichis in turn based on Hale [31]. The final result generalizes that ofSoumekh [8, eqs. (19) and (40)], and it does not involve seriesexpansion, nor small squint approximations.

Let us start from the impulse response of the bistatic SARacquisition. We simply extend the same model shown in [18] tothe bistatic case. Let us replace the axis in the model space,with the axis, that has a just a different origin. In this newreference, we will define the source and receiver location asand , respectively

(66)

The impulse response of a target located in is theflat-top hyperbola

(67)

where the suffix stands for “bistatic,” and and are thetarget-to-source and target-to-receiver distance, respectively

The summation leads to the DSR term.Let first transform such impulse response along time

(68)

where we have introduced the definition , andthe superscript “ ” to indicate the Fourier transform (FT) withrespect to that variable.

We then express each of the two factors in (68) as a superpo-sition of plane waves. We exploit the following inverse FT:

(69)

which is quite known in SAR community, as it is exploited toderive the transfer function for monostatic SAR (see [8] and [18]for a review). Notice that in (68), we have ignored complex con-stants, and we have neglected slowly varying amplitude terms,a quite reasonable assumption in SAR systems.

Combining (67) and (68), we get the decomposition of thereceived field into the wavenumber domains of the source andthe receiver in (70), shown at the bottom of the page.

Let then apply the change of coordinates (66) and, as a con-sequence, introducing the corresponding wavenumbers (in thisAppendix, we use instead of , for clarity)

The received field (70) is now expressed in thewavenumber domain and furthermore transformed asin (71), shown at the bottom of the next page.

We follow now the same approach underlying the DMOderivation: instead of trying to invert the forward model of the

(70)


bistatic SAR (71) (which would be quite complicated to derivein a closed form, and computationally expensive), we ratherconvert that model into an equivalent monostatic one. Themonostatic SAR acquisition is simply derived by substituting

in (68) and following the subsequent planewave expansion (70), that would keep only one term

(72)

and being the monostatic time and its companion angularvelocity ( ).

It is possible to show that under the change of variable

(73)

the following relation holds:

(74)

[it can be proven just by squaring both members and then ap-plying (73)].

The conversion from the bistatic field into the equivalentmonostatic one can then be accomplished by applying thechange of variable (73) to (71) and evaluating the output for

(75)

Notice that the term on the left is the monostatic field (72),which shares the same center of the bistatic and thesame wavenumber domain. However, in the change of variable

implied in (75), we have ignored the Jacobean

(76)

which would result in a amplitude term that changes slowly withtime for the bistatic SAR of interest.

The derivation of (75) is thus pretty general, and it is not lim-ited to small squint angles or small apertures. It involves an inte-gration in the domain of the 3-D transformed field, a changeof coordinates , and an inverse FT . Similarlyto the Stolt interpolation [16], the kernel implied by these stepsis range variant. Let us evaluate the kernel directly in the timedomain: we compute its (time-varying) impulse response by as-suming a bistatic field centered on some

(77)

whose FT is clearly

(78)

We then apply the change of variable and the integration oninvolved in (75) getting the expression of the kernel in the

domain

(79)Eventually, we can get an explicit form for this kernel by ex-ploiting the stationary phase to compute the integral. Let uswrite the phase term

(80)

The stationary phase approximation of (79) is then

sign (81)

to be computed in the stationary points , solutions of

(82)

These solutions can be expressed in the following form:

(83)

Inserting these values back into the phase (80), we get

(84)

(71)


Henceforth, if we ignore constant term, the time-varying kernelis

(85)

If we approximate the last term into the square root as

(86)

then (85) becomes a ( varying), time domain shift

(87)

which is the same result as in (35).

ACKNOWLEDGMENT

The authors wish to thank S. B. Fomel, who acknowledgedhimself as a revisor, for his accurate revision, for his help infixing typos, and for his kind suggestions and literature refer-ences.

REFERENCES

[1] S. Deregowski and F. Rocca, “Geometrical optics and wave theory ofconstant offset sections in layered media,” Geophys. Prospect., vol. 29,pp. 374–406, 1981.

[2] G. Bolondi, E. Loinger, and F. Rocca, “Offset continuation of seismicsections,” Geophys. Prospect., vol. 30, pp. 1045–1073, 1982.

[3] , “Offset continuation in theory and practice,” Geophys. Prospect.,vol. 32, pp. 813–828, 1984.

[4] D. Hale, “DMO processing,” Geophys. Reprint Ser., vol. 29, pp.374–406, 1995.

[5] D. Massonnet, “The interferometric cartwheel, a constellation of lowcost receiving satellites to produce radar images that can be coherentlycombined,” Int. J. Remote Sens., vol. 22, no. 12, pp. 2413–2430, 2001.

[6] A. Brenner and J. Ender, “Very wideband radar imaging with the air-borne SAR sensor PAMIR,” in Proc. IGARSS, Toulouse, France, July21–25, 2003, p. 3.

[7] M. Soumekh, “Bistatic synthetic aperture radar inversion with applica-tion in dynamic object imaging,” IEEE Trans. Signal Processing, vol.39, pp. 2045–2055, Sept. 1991.

[8] , “Wide bandwidth continuous wave monostatic/bistatic syntheticaperture radar imaging,” in Proc. Int. Conf. Image Processing, vol. 3,Oct. 4–7, 1998, pp. 361–365.

[9] O. Yilmaz and J. Claerbout, “Prestack partial migration,” Geophysics,vol. 45, pp. 374–406, 1980.

[10] Y. Ding and D. C. Munson, Jr, “A fast backprojection algorithm forbistatic SAR imaging,” in Proc. Int. Conf. Image Processing, vol. 2,Rochester, Oct. 22–25, 2002, pp. 441–444.

[11] J. F. Claerbout, Imaging the Earth Interior. Boston, MA: Blackwell,1985.

[12] A. Tarantola, Inverse Problems Theory. New York: Elsevier, 1987.[13] O. Loffeld, H. Nies, V. Peters, and S. Knedlik, “Models and useful rela-

tions for bistatic SAR processing,” in Proc. IGARSS, Toulouse, France,July 21–25, 2003, p. 3.

[14] J. Ender, “Signal theoretic aspects of bistatic SAR,” in Proc. IGARSS,Toulouse, France, July 21–25, 2003, p. 3.

[15] D. Massonnet, J. C. Souyris, and J. M. Gaudin, “Processing and geo-metric specificity of bistatic data,” in Proc. IGARSS, Toulouse, France,July 21–25, 2003, p. 3.

[16] R. Stolt, “Migration by Fourier transform,” Geophysics, vol. 43, pp.23–48, 1978.

[17] C. Cafforio, C. Prati, and F. Rocca, “SAR data focusing useing seismicmigration techniques,” IEEE Trans. Aerosp. Electron. Syst., vol. 27, pp.199–207, Mar. 1991.

[18] R. Bamler, “A comparison of range-Doppler and wave-number domainSAR focusing algorithms,” IEEE Trans. Geosci. Remote Sensing, vol.30, pp. 706–713, July 1992.

[19] J. L. Black, K. Schleicher, and L. Zhang, “True amplitude dip moveout,”Geophysics, vol. 58, pp. 47–66, 1993.

[20] C. D. Notfors and R. Godfrey, “Dip moveout in the frequencywavenumber domain,” Geophysics, vol. 52, pp. 1718–1721, 1987.

[21] S. Deregowski, “Dip move out and reflection point dispersal,” Geophys.Prospect., vol. 30, pp. 318–328, 1982.

[22] C. Liner, “General theory and comparative anatomy of dip moveout,”Geophysics, vol. 55, pp. 595–607, 1990.

[23] B. Zhou, I. M. Mason, and S. A. Greenhalg, “An accurate formulationof logstretch dip moveout in the frequency-wavenumber domain,” Geo-physics, vol. 61, pp. 815–820, 1996.

[24] A. Canning and G. H. F. Gardner, “Regularizing 3-D datasets withDMO,” Geophysics, vol. 61, pp. 1103–1114, 1996.

[25] S. Fomel, “Theory of differential offset continuation,” Geophysics, vol.68, pp. 718–732, 2003.

[26] B. Biondi and S. Ronen, “Dip moveout in shot profiles,” Geophysics,vol. 52, pp. 1473–1482, 1987.

[27] J. Ronen, “Wave equation trace interpolation,” Geophysics, vol. 52, pp.973–984, 1987.

[28] F. Gatelli, A. M. Guarnieri, F. Parizzi, P. Pasquali, C. Prati, and F. Rocca,“The wavenumber shift in SAR interferometry,” IEEE Trans. Geosci.Remote Sensing, vol. 32, pp. 855–865, July 1994.

[29] S. Deregowski, “What is DMO,” First Break, vol. 4, no. 7, pp. 7–24,1986.

[30] A. M. Popovici, “Prestack phase-shift migration for separate offsets,”in Stanford Exploration Project Report. Stanford, CA: Stanford Univ.,1993, vol. 79, pp. 115–127.

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Davide D’Aria was born in Bergamo, Italy, onNovember 12, 1975. He received the telecommu-nication engineering degree from Politecnico diMilano, Milan, Italy, in 2000, with a thesis on SARinterferometry.

In 2002, he was with the SAR processing groupwithin the Electronic and Information Department,Politecnico di Milano, where he worked on SAR fo-cusing techniques, SAR-SPOT superresolution algo-rithm definition, bistatic SAR processing, and activeantennas calibration. He is currently with ARESYS,

a Politecnico di Milano spin-off company, where his activity areas are efficientfocusing processor software development, SAR commercial products valida-tion, and third-party processor testing.

Andrea Monti Guarnieri was born in Milan, Italy,on February 9, 1962. He received the laurea degreein electronic engineering from the Politecnico di Mi-lano, in 1988.

Since 1988, he has been with the Dipartimento diElettrotecnica ed Elettronica, Politecnico di Milano,where he joined the Digital Signal Processing team,and he is currently an Associate Professor. He teachescourses on signal theory, signals and systems, andradar theory and techniques. His research interestsconcern digital signal processing, mainly in the field

of synthetic aperture radar signal processing. Since 1987, he has authored morethan 50 scientific publications in the field of synthetic aperture radar.

Dr. Monti Guarnieri was awarded the Symposium Paper Award at theIGARSS in 1989.


Fabio Rocca received the Dottore degree (cumlaude) in ingegneria elettronica from the Politecnicodi Milano (POLIMI), Milan, Italy, in 1962.

He is currently a Professor of digital signal pro-cessing. He has been with the Department of Elec-tronic Engineering, POLIMI, since 1962, first as aTeaching Assistant, an Associate Professor, and Pro-fessor of radiotechniques. He visited the System Sci-ences Department, University of California, Los An-geles, from 1967 to 1968, and then the Department ofGeophysics, Stanford University, Stanford, CA, from

1978 to 1979, 1981, 1983, and 1986. His research work has been devoted mainlyto video coding, seismic data processing, and synthetic aperture radar.

Dr. Rocca was Coordinator of the first European Economic Community re-search program in geosciences. From 1975 to 1978, he was Chairman of the De-partment of Electronic Engineering, POLIMI, and was then on the UniversityBoard (Commissione d’Ateneo, 1980–1993). He was President of the Osserva-torio Geofisico Sperimentale, a National Institute for Research in Geophysics,from 1982 to 1983. He is a Member of the Scientific Councils of the InstitutFrancais du Petrole, of the Istituto Nazionale di Oceanografia e di GeofisicaSperimentale (OGS), and Scientific Advisory Groups for the European SpaceAgency. He is an Associate Editor of the journals Signal Processing and SeismicExploration. He is Past President of the European Association of ExplorationGeophysicists. In 1979, he received the Honeywell International Award, in 1989and 1999 the IGARSS Symposium Prize Paper Awards, in 1990, the EAGESchlumberger Award, in 1995, the Italgas Telecommunications Award, in 1998,the SEG Special Commendation Award, in 1999, the Eduard Rhein FoundationTechnology Award for motion compensated video coding (divided with Dr. L.Chiariglione), and in 2001, the Honoris Causa Doctorate in Geophysics fromthe Institut Polytechnique de Lorraine. He is an Honorary Member of the So-ciety of Exploration Geophysicists (1989) and the EAGE (1998).

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