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Focusing Bistatic Synthetic ApertureRadar using Dip Move Out

D. D’Aria, A. Monti Guarnieri, F. RoccaDipartimento di Elettronica e Informazione - Politecnico di Milano

Piazza Leonardo Da Vinci, 32 - 20133 Milano - Italytel. +390223993446, fax. +390223993413, e-mail: monti@elet.polimi.it

Abstract— The appearance of new Synthetic Aperture Radaracquisition techniques based on opportunity sources enhancesinterest in bistatic geometries. In seismic data acquisition, eachsource is currently accompanied by up to 10000 receivers and, inthe last two decades, the bistatic geometry has been carefullystudied 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 namedDip Move Out or DMO. In essence, the elliptical locus ofthe reflectors corresponding to a spike in the bistatic survey,is forward modeled as if observed in a monostatic one. Theoutcome of the model, the so calledsmile, is a short operator,slowly time varying but space stationary. To transform a bistaticsurvey into a monostatic one, it is enough to convolve the initialdata set with this smile. Based on the well known similaritybetween seismic and SAR surveys, DMO is first described inits simple geometric understanding, and then used in the SARcase. The same processing that is being used for MovementCompensation can be applied to the bistatic to monostaticsurvey transformation. Synthetic examples are also provided.

I. I NTRODUCTION

In this contribution, we shall try to pass to the radarcommunity the understanding of bistatic surveys in terms ofthe so called Dip Move Out or DMO, well known in seismicssince the early eighties; in doing that, we will use materialpublished by seismic professionals in the eighties and nineties[1], [2], [3], [4], but also using as long as possible, radarterminology besides the seismic one.

The renovated interest in bistatic radar surveys is due toseveral new acquisition techniques, where the illuminatorcan be an opportunity one. Typically, in the cartwheel [5]approach, three to four ”micro receivers” are orbited close toa satellite SAR system, for instance trailing it at a distance ofa few tens of kilometers. Their orbit is not exactly identical,so that they appear to describe an ellipse to an observerthat tracks the transmitter. For a sizeable part of the totaltime, a couple of receivers is in the proper position forinterferometry, be it along or across track. The processingto be done for the focusing of the data received by thesemicroreceivers is close but not identical to that correspondentto the zero offset (distance between source and receiver)

acquisition, i.e. where the receiver is positioned very closeto the transmitter.

There is never such a case of zero offset, since the wavepropagates in a small but non zero time, to return to thesource after it has moved ofvsat2r/c i.e. a few meters,in the usual satellite situations. So, if the minimum offsetin the satellite case is say a few meters, the maximumenvisaged in the case of the cartwheel could be severaltens of kilometers. Greater offsets are made unlikely bythe decreasing backscatter amplitude in directions furtherapart from the specular one. Other situations correspond tothe case of an orbiting illuminator, say a GPS transmitter,and a ground based receiver. The case when the offset istime varying should be studied specifically. However, wenotice that the change in offset should be considered duringthe time that the target dwells within the footprint of thetransmitter’s antenna (or the receiver’s if it is narrower), andthis time might not be very large. Typically, the motion ofthe cartwheel during a footprint is very small and will beneglected in our analysis. Thus we will be able to study thesituation as if the offset was time stationary, even if this isnot exactly the case.

So we consider stationary situations, i.e. where the sourcedescribes a straight line, and the receiver follows the sourceat a given offset. The results come from the Born approx-imation, where multiple reflections are systematically ne-glected; in other words, we are supposing that each reflectoracts as a point scatterer, independently of the others. Inseismics, where the average reflection coefficient is a fewper cent (changes in rocks velocities or densities are neverthat great, unless at the air- or sea-terrain interface), this istotally acceptable. Not so in the electromagnetic case, wheremany strong scatterers exist, that often have a reflectioncoefficient close to one, and multiple reflections within orwithout the same object are not negligible at all. A splendidpictorial description of such a situation has been recentlymade available [6] by the results of a very high (0.1m)resolution radar system (PAMIR) where the physics of thescatterers is well understandable and the gigantic dynamicsof the electromagnetic reflectance in presence of conductorsis well visible; no such a thing exists in seismics, where

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the world is transparent to the elastic waves that cross it.It should be well understood that the entire concept of thepossibility of transforming a bistatic survey into a monostaticone is meaningful ONLY IN THE CASE of small reflectioncoefficients. Otherwise, one could concoct counterexamples,where objects lose or gain visibility depending on the bistaticangle of observation, entering into 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 termsof the cross correlation with the hodograph of the data(i.e. their Doppler history). Following Yilmaz and Claerbout[9] who first approached the problem in the open seismicliterature, in the sequel this hodograph will be referredto as the ”Double Square Root equation” or DSR fromthe two square roots that appear in the expression of thetravel path. These two square roots reappear in other formsmoving to the wavenumber domain [1], [7]. However, thekey advantage of the DMO technique, namely the possibilityof decomposing the complex, time varying operator resultingfrom the cross correlation with the phase history of the DSR,into the cascade of the usual monostatic processing plusthe application of a time varying, but short operator (i.e.the DMO operator) is not mentioned in Soumekh [7], evenif the DMO operator shows up in a line of the derivation(eq. 19). In other papers, the same author has publishedmore considerations on the DSR, be it in the space or inthe wavenumber domain. Munson [10] has discussed theproblem in terms of the backprojection, that is the adjointoperator. Again following Claerbout [11], we can refer topull or pushoperators 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 isn’t, we should usean 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. Moreauthors [13], [14], [15] are now studying the problem withinthe radar context, and therefore we think it could be usefulto reanalyze, using also the radar geometry, the problem ofconstant offset acquisitions.

After a short discussion of the complete push and pulloperators (the ellipse and the DSR) we will introduce theconcept of DMO in sections IV and V, first in the spacedomain and then in the wavenumber domain. The analysisof its behavior with radar frequencies and geometries will bediscussed in section VIII and an example of the results of theprocessing will be given in section XII, together with a simplerecipe to transform a code for Motion Compensation usingsubapertures into a code for bistatic focusing. Both types of

codes share the time varying character as well as the natureof being a prequel to a monostatic focusing procedure, be itanyone of choice.

II. FOCUSING MONOSTATIC SURVEYS

To get familiar with the way of thinking of the geophysi-cists, we first consider the well known monostatic surveys.We consider first the data space, correspondent to the domainwhere the data are acquired. As a function of the abscissayof the source, coincident with that of the receiver, we measurereflections at (monostatic) timestm (y). We distinguish thedata (or signal) spaceD(y, tm), from the model space wherethe scatterers are, their position being a function of theabscissa y again and of the slant rangeζ, distance fromthe trajectory of the source. If we have one spike in the dataspace, at timetm = t0, when the illuminator and receiveris at the abscissa0, then the scatterers are located along thecircle:

y2 + ζ2 =c2t204

(1)

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 an hyperbola:

y2 − c2t2m4

= ζ2 (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 constantc, then we know that the only possiblepositions of the reflectors in the model space are alongthe circle (1). If we had only one scatterer with coordinates(y = 0; ζ) then the echoes would be located in the dataspace along the hyperbola in the coordinates(y, tm) (2). So,if we have a single scatterer, we measure one hyperbola; ifwe superpose the effects due to the many scatterers that makethe circle, we have to superpose the correspondent hyperbolasin the data space and we get back the initial spike, as shownin Fig. 1.

Viceversa, suppose we start with a single scatterer in themodel space; we see an hyperbola (only one, now) in themodel space, and then we expand each measurement intothe corresponding circle in the model space. Again, all thecircles will superpose 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 (thepull approach) we collect all the echoes relative to a givenpoint scatterer by averaging the amplitudes in the data spacealong the correspondent hyperbolas (2) (focusing by crosscorrelation). This is shown in Figs 1 and 2 that demonstratethe reconstruction of the pulse. These considerations arepurely kinematical; it is also well known that, unless theproper weighting along circles or hyperbolas is adopted, there

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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 composethe circle, an hyperbola corresponds in the data set. The sum of all thesehyperbolas recreates the observed spike.

Fig. 2. Focusing, as seen in the model space. A point scatterer generatesan hyperbola in the monostatic data set. The circles that correspond to thespikes that compose the measured hyperbola superpose on the initial pointscatterer.

is a residual defocusing that should be dealt with the so calledrho filtering [11].

The key point is that the cross correlation with the timevarying hyperbola, operation that could appear computation-ally heavy, is made cheap ad simple by well known focusingtechniques like the Stolt interpolation [16] [17] [18].

III. F OCUSINGBISTATIC SURVEYS

To focus bistatic surveys we will use the same procedureas for the monostatic approach, supposing the reflectors tobe infinitesimally weak, and then using the superpositionprinciple. So now let us consider a bistatic survey (thegeometry is indicated in Fig. 3) and let us indicate withs, rthe positions of the source and receiver.

We suppose that the receiver trails the source at a constantoffset 2h:

h =s− r

2(3)

The midpoint y between source and receiver is indicatedagain with:

s rh

Equivalentm

onostatic

y

Fig. 3. Geometry of the bistatic system.

y =s + r

2(4)

So, if h = 0 we are back to the monostatic case. Anybistatic data setD(y, tb), where y is the azimuth andtb the bistatic arrival time, can be decomposed, using thesuperposition principle, in a superposition of pulses. Withoutloss of generality, as before, we suppose that we measure onlya reflected pulse, at the timetb and whens = − r = h/2 i.e.at the positiony = 0 . We suppose again that the velocityof the medium is a constantc. Then, we can say that thereflectors that created this echo are spread along an ellipsein the model space. This is the 2D physical space that hasfor horizontal coordinatey, the same as for the data space,but that has as vertical coordinate the spatial coordinateζ(slant range) and not the time of the arrival as in thedataspace. This ellipse has foci in the positionss, r; the lengthof its horizontal semi axis isa:

a =ctb2

(5)

and that of the vertical semi axis isb :

b =√

a2 − h2 (6)

In fact, the total travel time from any point of this ellipseto source and receiver adds up totb. Any other point in themodel space corresponds to a different value of total traveltime. Since we are supposing that the elements of the ellipseare point scatterers, it is evident that this analysis is purelykinematical and not dynamical; in other words we are notattempting to appreciate amplitudes. Thus, we are not tryingto calculate the reflectivity of the elements of the ellipse,that needs not to be constant. Besides, the thickness of theellipse is also not considered and the consequent slant rangeresolution; however, this is typical of the stationary phaseapproach, and correspondingly of the optical ray theory.

We note in passing that a lot of attention has been ded-icated, in the seismic literature, to the amplitudes problem,

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Fig. 4. A spike in the bistatic data set corresponds to an ellipse in themodel; then, to each scatterer of the many that compose the ellipse, a flattop hyperbola corresponds in the data set. The sum of all these flat tophyperbolas recreates the observed spike.

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

[4], [19]; we believe that this problem is still premature forradar.

As observed, we indicated witha the horizontal axis (thelongest, along azimuth) of the ellipse and withb its shortestaxis, vertical, along slant range. Using the current coordinatesη, ζ in the model space, the equation of the ellipse is:(

ζ

b

)2

+(η

a

)2

= 1 (7)

As the ellipse is the correspondent of the circle forh 6= 0,in correspondence of a single point scatterer in the modelspace, we have aflat top hyperbolain the data space insteadof the hyperbola seen before. In fact, if we have a singlescatterer at abscissa 0 and at slant rangeζ in the model space,in the data space we will see reflections at the times givenby the so called Double Square Root (DSR) equation [11](see Fig. 5):√

(y − h− η)2 + ζ2 +√

(y + h− η)2 + ζ2 = ctb (8)

The focusing comes from the superposition of effects; in

Fig. 4, we see the effects of a spike in the data space. Thereis an ellipse in the model space, and each of its pointscorresponds to a different flat top hyperbola, again in thedata space. Their superposition

confirms that there was just the initial spike in the dataspace. In Fig. 5 we see the effects of a spike in the modelspace. Thus there is only one flat top hyperbola in the modelspace, and each of its points corresponds to a different ellipse,in the model space. Their superposition recreates the initialspike in the model space. In both cases we see that the flattop hyperbola is theinverse operatorto the ellipse, as thehyperbola was the inverse to the circle. This is good newsand bad news.

Good news because we know now how to focus bistaticsurveys; it is enough (pull) to sum along flat top hyperbolas inthe data space (correlation in data space) or (push) to spreadeach sample in the model space along its correspondingellipse (backprojection).

Bad news because these techniques are expensive and wecannot reuse the focusing machinery developed for mono-static surveys (spreading along circles or summing alonghyperbolas).

IV. T HE SMILE

Let us assume the same semi-elliptical distribution ofreflectors in the model space that causes the spike in thedata space shown in Fig. 4. In order to reuse the monostatictechniques, we need to express the same bistatic data setas the superposition of many monostatic datasets, manycircles centered in different position along azimuth, and withdifferent radii (delays), as shown in Fig. 6. We may thinkas a semielliptical reflector imaged by different monostaticsurveys. Each of these survey will show arrivals (specularreflections on the points of the semiellipse) in an interval(−ymax ÷ ymax) of positions of the source (coinciding withthe receiver). If the source is outside this interval, there isno specular reflection. If the abscissa of the source is insidethis interval, the reflections will arrive at the times when theexpanding circular wave centered on the source location atthe abscissay (within the interval) is tangent to the ellipseand thus is mirrored back to the source. Let us refer to the 2Dgeometry sketched in Fig. 7.To find the arrival times of thespecular reflections on the ellipse, we indicate withη, ζ thecoordinates either of the points of the circles correspondingto the monostatic survey or of the ellipse; indicating withtmthe monostatic traveltimes and thus withd = ctm/2 the radiiof the circles, we have:

(η − y)2 + ζ2 = d2; tm =2d

c(9)

The equation of the location of the reflectors in the modelspace, i.e. the ellipse, is again:(

ζ

b

)2

+(η

a

)2

= 1 (10)

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Fig. 6. The semielliptic reflector sketched as marked dotted line, thatcorresponds to a spike in a bistatic data set, is observed by a set of monostaticsurveys (circles). Dots indicate the locations of the centers of the circles,corresponding to the time of arrivals of reflections from the elliptic target.

θ

ya

b

2a=ctb

s rctm(θ)/2

2b=ctm(θ=0)=ct0

Slant range

Azimut

ζ

Fig. 7. Geometry for the derivation of the DMO operator. The upper plotallows the calculation of the proper monostatic delay,tm, for each azimuth,y, in the model space. These delays are then applied in the data space (y, tm),as shown in the lower figure. The locus of these delays is the smile operator.

Imposing a common tangent to the circles and ellipse torepresent the specular reflection:

q =dζ

dη= tan θ (11)

We equate the dummy variablesη, ζ (the coordinates ofthe backscattering point on the circle and the ellipse) andtheir derivativeq (the tangent of the angle of the elementarymirror)

(η − y) + ζq = 0 (12)

ζq +ηb2

a2= 0 → ηb2

a2= η − y (13)

Then, substituting back for the dummy variablesη, ζ we getthe monostatic arrival timestm (y) to and from the ellipseas a function of the position of the illuminator. If we nowplot the arrival times as a function of the positiony of theilluminator, we get a line in the data space as Fig. 8 shows,that pertains to an ellipse (yes, but another one, not that in

the model space we started with) with equation (in thedataspace!):

t2mt20

+y2

h2= 1 (14)

This ellipse bottoms at time:

t0 =2b

c=

√t2b −

4h2

c2(15)

for y = 0, and touches they axis at the abscissas±h. Notall the ellipse pertains to the locus, since forq = ±∞ wehaveζ = 0; η = ±a ; the locus covers only a small part ofthe ellipse (and has thus been named ”smile” for its peculiarshape). The horizontal extension of the locus is the interval(−ymax ÷ ymax):

ymax ≤2h2

ctb= h

h√h2 + b2

< h (16)

Within this extension, and since the offseth is in general

S Rζ

y

a

b

y(θ3)y(θ2)y(θ1)

tm(θ1)

S

tm

yR

ymax=h2/a

tm(θ2)tm(θ3)

2b/ctm(θ3)tm(θ1)

Fig. 8. Geometry for deriving the smile in the azimuth time data domain.

much smaller than the maximum monostatic slant rangeb, it is in general possible to expand the square root andapproximate:

tm = t0

√1− y2

h2∼ t0 −

y2t02h2

(17)

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

t0 =

√t2b −

4h2

c2∼ tb −

2h2

c2tb(18)

in other words, a monostatic survey would see any scattererat earlier times.

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

Then, instead of focusing the ellipse with a bistatic expen-sive technique, using a cheap monostatic technique we canfocus the superposition of circles, provided that we replaceeach sample of the bistatic survey with the correspondentsmile to transform the bistatic data set into a monostaticone. Thus, with the convolution with the smile, we have

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Fig. 9. The smile is obtained from the envelope of the hyperbolas thatcorrespond, in the data space, to each point scatterer that composes theellipse, in the model space.

transformed the flat top hyperbola into the usual hyperbola(Fig. 10). With respect to the complexities of using the fullyextended flat top hyperbola to focus, we see that there is theadvantage that the smile is a short operator. We still have thedisadvantage that the smile is time varying. We will see thatthis drawback can be overcome with a logarithmic stretch[2], [20].

Fig. 10. A single scatterer in the model space generates a flat top hyperbolain the bistatic data set. If, instead of each point of the flat top hyperbolathe proper smile is positioned in the data space, their envelope will be thevery same hyperbola that corresponds to the initial point scatterer, but in amonostatic survey.

V. WAVENUMBER DOMAIN FOCUSING : SLOPES OF THE

TANGENTS TO THE SMILE

Let us now see how to convolve with the smile, and inparticular, how to take into account the dip of the reflector orthe squint of the beam. In this section we indicate withtb (θ)the bistatic travel time correspondent to the illumination ofa reflector slanting of the angleθ; in radar terminology,θis the squint angle of amonostatic survey. If we have onlyhorizontal reflectors(θ = 0), i.e. for zero squint, to transforma bistatic survey that sees an arrival at timetb (θ = 0) into

a monostatic one, it is enough to anticipate the data in thedata space of the time:

tm(θ = 0) =

√t2b(θ = 0)− 4h2

c2(19)

Suppose now we had a reflector dipping by the angleθ(hence, not a point scatterer), the same results come if thebeam is squinted fore or aft of the same angleθ. In this caseof non zeroθ, the offset entails a smaller delay (Dip MoveOut or DMO in seismic terminology) as seen from Fig. 11.We indicate withtm (θ) the monostatic traveltime:

R

ctb

ctm

h y

r

s

S bisector

ctm

X

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

tb (θ) =tm (θ)cos β

= tm (θ)√

1 + tan2 β (20)

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

sin(900 − θ)ctb (θ)

=sinβ

2h⇒ cos2 β = 1− 4h2

(ctb (θ))2cos2 θ

that leads to the following the relation:

1 + tan2 β = 1− (ctb (θ))2

(ctb (θ))2 − 4h2 cos2 θ(21)

We then combine expressions (20) and (21):

t2b (θ) = t2m (θ)(1 + tan2 β

)= sin2 θ

(ctb (θ))2

(ctb (θ))2 − 4h2 cos2 θ

1 = t2m (θ)1

(ctb (θ))2 − 4h2 cos2 θ(22)

This expression leads to the same smile operator, just derivedin the previous section:

t2b (θ) = t2m (θ) +4h2

c2cos2 θ (23)

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Notice that the total delay of a bistatic survey with respectto that of a monostatic survey is maximal for zero squint(θ = 0)

tb(θ = 0) =

√t2m +

4h2

c2(24)

and goes to zero,tb(θ = 900) = tm(θ = 900) when thesquint is θ = 900. Thus, the smile can also be seen as anegative delay as a function of the squint of the beam,θ,i.e. the Doppler frequency in radar terminology (figure 12).The same results can be got directly from the smile equation,with some algebra. In fact, it can be shown that the tangentto the smile that has slope:

dtmdy

=2 sin θ

c(25)

intersects they = 0 line for

tm (θ) =

√t2b −

4h2 cos2 θ

c2(26)

Fig. 12. A smile decomposed in the envelope of its tangents; instead ofcrossing in a point, these lines are delayed (moved out) differently dependingon their dip (Dip Move Out). The external tangents pass through the dot thatindicates the bistatic travel time. In fact,900 squint entails no delay changebetween the monostatic and the bistatic cases.

Therefore, instead of convolving with the smile, we canapply different negative delays as a function of the squintθ.This negative delay is:

tθ =

√t2b −

4h2 cos2 θ

c2∼ tb −

2h2

c2tb+

2h2 sin2 θ

c2tb(27)

tDMO =2h2 cos2 θ

c2tb(28)

Equation (28) agrees with equation (14): however, in onecase (14) the delay is seen as a function of the position ofthe source in the monostatic survey; in the other case (28)the delay is a function of the squint of the beam. Thus, thesmile is seen as the envelope of its tangents, each of themcorresponding to a different squint, i.e. a different Dopplerfrequency, i.e. a reflector with dipθ (in a short time window,to avoid the effects of the time variance). To convolve with

the smile, we can then decompose the2D input data setD (y, tb) into several1D subapertures in the Doppler domain,apply to each subset a time varying negative delay, and thenrecombine. This is easily done in the horizontal wavenumberk and time domain, or even better in the wavenumber-frequency domain, if we learn how to deal with the timevariance of the smile.

Let us express with equations the concepts that we haveseen in this section, up to now. Let us indicate withk themonostatic azimuth wavenumber, and express its relationwith the squint angleθ, slope of the reflector. The monostaticDoppler frequency∆fD is:

∆fD =2vsat

λsin θ =

2fvsat

csin θ (29)

The squint angleθ is thus related to the Doppler frequencydomain variablesk − ω :

sin θ =kc

2ω(30)

2π∆fD

vsat= k (31)

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

sin θ =c

2dtmdy

=c∆fD

2f0vsat(32)

The delay as a function of the Doppler frequency is then:

t2b = t2m+4h2

c2(1−sin2 θ) = t2m+

4h2

c2

[1−

(kc

2ω

)2]

(33)

λ∆fD

2v=

kc

2ω=

c

2dtmdx

= sin θ (34)

Finally, solely by applying the negative delay as a functionof the Doppler frequency:

tm =

√t2b −

4h2

c2+(

h∆fD

f0v

)2

(35)

we transform the bistatic data set into a monostatic one.

VI. T HE LOGARITHMIC STRETCH

We still have to cope with the non stationary behavior ofthe smile with time. The easy way is to apply the so calledlogarithmic stretch to the input data. We start from equation(28) to see that if we pose:

τ = T logtbT

(36)

thendτ

dtb=

T

tb→ ∆τ

T=

∆tbtb

(37)

So, instead of a time varying delay:

2h2 cos2 θ

c2tb(38)

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we will apply the one

2h2 cos2 θ

c2T(39)

For T we can take the minimum two way travel time of thesurvey. Indicating withΩ the domain conjugate to that ofthe variableτ, we can now operate in theΩ, k domain byapplying the operator:

exp[−jΩ

2h2

c2T

(1− k2c2

4Ω2

)](40)

nilpotent forh = 0, as it should [2], [20].Clearly, (40) is an approximation, it can be shown that the

exact log-stretch phase shift is:

exp jΩT

2

[√1 + ε2 − 1− ln

(1 +

√1 + ε2

2

)](41)

whereε =2kh

ΩT

Equation (41) was derived in different ways by Liner [22],Zhou et al. [23], and Canning and Gardner [24]. In reference[25], it is derived by solving the offset continuation partialdifferential equation. The equation solution suggests an am-plitude factor in addition to the phase shift but it can beneglected if amplitudes

are not important for data processing. The implementationof the log-stretch operator requires that data are resampled,in the fast-time direction, according to (37), however, weassume irrelvant the cost of this monodimensional operatorin the budget of the whole 2D bistatic focusing.

VII. O FFSET CONTINUATION: A PARTIAL DIFFERENTIAL

EQUATION

It can be of interest to observe that there is anothertechnique to focus bistatic data, named ”offset continuation”.A partial differential equation can be found, that links thedataD (t, y, h) measured at different offsets, but related tothe same model in the model space [2], [25].

∂2D

∂y2− t

h

∂2D

∂t∂h=

∂2D

∂h2(42)

The advantage of this approach comes when differentreceivers have to be combined together into a single image.Then, the above equation acts as a train going towards thefinal station ath = 0, leaving fromh = hmax . New dataare loaded on the train (stacked) whenever the offset is thecorresponding one. In other words, let us start the integrationof the equation using as initial condition the data measuredat h = hmax. Then, we use the equation to continue the datato a smaller value ofh, h1, stack in the data from the secondreceiver that hash = h1, and so on, until the monostatic dataare also stacked in, when finallyh = 0. If the receivers arein the thousands, this technique comes in handy.

VIII. R ADAR GEOMETRIES

In the previous sections we have decomposed an entireellipse into the superposition (the envelope, that is) of numer-ous circles, transforming a bistatic survey into a monostaticone. In the radar case, and even more if the radar is spaceborne, the part of the reflectors ellipse in the model space thatis illuminated by the source is limited since the beamwidthof the transmitting antenna is in general small. This is donein order to keep the Signal to Noise Ratio high enough, andthen to prefilter the data to remove the spatial alias and limitthe spatial sampling rate to that correspondent to the imageresolution.

We should consider the fact that, if the Doppler spectrumis limited by the antenna, the same should happen to theazimuthal spectrum of the smile. It is not efficient to convolvetwo data sets, unless the spectra have the same support,especially if there are risks of alias. Now, let us indicatewith k0 the central Doppler frequency. This need not to bezero, especially if the monostatic SAR is pointed broadside,towards zero Doppler, that is. The monostatic survey corre-spondent to the bistatic survey, due to the offset2h of thereceiver, is pointed at Doppler;

k0c

2ω= sin θ0 = −h

d(43)

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

proximation (28) holds,in this transformation from bistaticto monostatic surveys, there are no change of the Dopplerfrequency, but only phase shifts. Thus, the interferometriccharacteristics of the data are not modified.

If the Doppler band of the bistatic survey is limited bythe footprint F of the illuminator, indicating with∆y theresolution along azimuth:

F =λd

2∆y(44)

The maximum and minimum squint anglesθmax, θmin are

θmax,min = θ0 ±λ

4∆y(45)

and consequently we can determine the minimum and max-imum azimuth wavenumbers. Depending onh, we mightvery well have that the bistatic survey has a spectrum totallydisjoint from that of the monostatic survey.

From this comes the rationale of the cartwheel, since inthat case the relative offsets are kept systematically smallenough to ensure the spectral overlap, at least between thedata recovered by the elements of the cartwheel. To move tothe monostatic survey, we have to apply Doppler and timedependent negative delays (the smile). The maximum delaywithin the spectral support of the data, limited by the onboard antenna is:

tsmile =y2t02h2

< tmax smile =y2max smilet0

2h2(46)

9

c

2dtsmile

dy<=

ct02

ymax smile

h2= d

ymax smile

h2=

λ

2∆y(47)

ymax smile =λh2

2d∆y(48)

tmax smile =t0

2h2

(λh2

2d∆y

)2

=t08

(λh

d∆y

)2

(49)

ctmax smile =λ2h2

4d∆y2(50)

In a typical case (L band, 4 m azimuth resolution) we getthe reassuring figure:

ctmax smile =(0.24)2

4× 600000200002

16= 0.6 m (51)

In the extreme case of 0.1m resolution and X band

ctmax smile =λ2h2

4d∆y2=

200002

4× 600000

(0.030.1

)2

= 15 (52)

and therefore 150 resolution cells. This number grows furtherif the main beam is squinted away from the bistatic receiver.This proves that with the expected values, DMO will not bea very taxing radar processing.

IX. N ON STATIONARY SURVEYS

Up to now we supposed that the offset between source andreceiver was constant; what if it changes during the survey?Obviously, we need to consider only the changes of theposition of the receiver while the same target stays within thefootprint of the source antenna, so that some approximationcan be accepted. We consider first the case of a stationaryilluminator. In this case, studied by the geophysicists as theShot DMO [26], we have that a spike measured at any givenlocation and time corresponds to scatterers located along anellipse that depends on the positions of source and receiveras seen previously. In this case, the non stationarity of thepulse response, neither in short nor in long times, excludesthe possibility of a convolutional solution. The data consistof delayed one way hyperbolas, each one correspondent to asingle scatterer. The solution proposed by the geophysicists istailored to be cast back into monostatic surveys, to simplifythe data handling. Again, the general use of the Born approx-imation is still useful. Let us consider each pulse emitted bythe source positioned in the locations (tl) (bold, since it isa vector, as it is any position say on a plane, and a functionof the ”long” acquisition timetl). It is backscattered by thereceiver positioned in

the locationr (tl) and arrives at the timetb (tl). The locusof the scatterers is a rotational ellipsoid with foci ins, r andmain axis with lengthctb/2. The application of the sameDMO principle shows that a monostatic survey on this dataset corresponds to a smile (a line in the 3D space spanned by

s, r, tb) laying in the vertical plane containings (tl) , r (tl)that bottoms in:

t0 =

√(ctb2

)2

− h2; |s− r| = 2h (53)

More pulses in other positionss (tl) , r (tl) will cor-respond to more smiles in the correspondent planes. Thepositions of the equivalent monostatic sourcesy (tl) are notalong a line anymore, but on a surface.

y (tl) =s (tl) + r (tl)

2(54)

This entails the existence of non zero baselines and thereforeinterferometric effects. Combining the Dip Moveout tech-nique with that of motion compensation, it should still bepossible to transform this data set into that acquired bya monostatic survey, apart from the unavoidable interfer-ometric effects. However, the problem becomes longer todescribe, and we will postpone its analysis to another paper.The problem of 3D DMO has indeed long been studied inseismics and can be extended to this case, when useful [4] .

X. STACKING SMILES

A further advantage of multistatic surveys comes fromthe possibility of combining the data to improve the overallspatial resolution, limiting at the same time the overalldata rate to efficient values. It is well known that a timeseries sampled at intervalT can be subsampledM times toobtainM different time subseries that can be recombined toreproduce the original one. Necessary condition for that isthe ”linear independence” of the subseries. In other words,all parts of the original spectrum should be recoverable andthe alias should be removable. The same concept applies tomultistatic surveys: it is reasonable to expect that say usingthree receivers instead of one, it should be possible to haveeach of them using a PRF reduced at best to 1/3 to recombinethe data to recover the original. This may happen say if themidpoints of the three surveys are staggered in space, or ifthe effect of filtering due to the smiles is properly used [27],[28]. Obviously, to avoid spectral holes, the offsets shouldbe properly positioned. However, it could be an additionaladvantage of the cartwheel geometry as it was observed fromthe beginning too [5], supposing that each platform observeda disjoint part of the entire Doppler spectrum. It is interesting,in the case of DMO, remark the effects of a point scattererin the model space. The superposition of several coregisteredbistatic surveys transformed into the monostatic one throughDMO, corresponds to a monostatic data set where severalsmiles are stacked together. Their envelope results to be partof the hyperbola that corresponds to the initial point scatterer[29].

10

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 materialinvestigated. The problem is very important in geophysics,since from the amplitude of the reflections very valuableinformation on the structure of the reservoir can be got,say determining whether there is a oil gas contact or not.The situation is not that mature for radar data, where alsothe much greater dynamic of the data would make such ananalysis much more difficult. The best known reference forthat is the paper by Black [19] also available in [4].

XII. I MPLEMENTATION FOR MICROWAVE SAR

Let us come to the case of microwave SAR, where dataare down converted, time domain windowed and sampled.If we define tb the time of thedown converteddata andωits correspondent wavenumber, we need to replaceω0 + ωin place ofω in the implementation of the smile operator.This notation is consistent with the appendix: the operator in(35) and (85) is to be applied to each single contribute in therange compressed data,δ(t− tb), to convert the signal intothe one achieved through a monostatic SAR. The resultingthe time-varying shift1 is thus (85):

H(ω, k; tb) =

= exp

(j (ω + ω0) tb

(1−

√1− 4h2 cos2 θ

t2bc2

))

' exp(

j (ω + ω0)2h2 cos2 θ

tbc2

)(55)

where

cos2 θ = 1− k2c2

4ω2∼ 1− k2c2

4ω20

(56)

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

H(ω, k; tb) ' exp

(jωtb

(1−

√1− 4h2 cos2 θ

t2bc2

))(57)

× exp

(jω0tb

(1−

√1− 4h2 cos2 θ

t2bc2

))(58)

The first operator, (57), isω-varying, hence implementsa migration (in the received field): it corresponds to theazimuth-wavenumber dependent delay of the smile (50). The

1According to the notation in the appendix, we useΩ to define thedomain conjugate to the time of the converted monostatic field,tmon. Thistime differs from the bistatic time of the raw data,tbis, due to the stretchimplemented by the smile. However this stretch is so small in the durationof the echo that can be neglected, hence we will useΩ andω with the samemeaning.

second operator, (58), is still the smile, but as a pure phaseterm, due to data demodulation. This operator is not presentin the geophysical formulation. However, the small fractionalbandwidth of usual SAR makes the smile-phase (58) thedominant term.

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

Themigration operator(57) is slow time-varying, and canbe implemented block-wise (the smile is actually a very shortoperator), as a shift in the range-Doppler domain (or a linearphase in theω, k domain). As an option, the operator couldbe made time-invariant by exploiting the logarithmic stretchin section VI. In most cases, it is sufficient to implement thedelay at the nominal squint angle,θ0

Hm(ω; tb) ' exp(

jω2h2 cos2 θ0

tbc2

)(59)

The positive phase slope remind us that the operator actuallyanticipates data depending on the wavenumber, as alreadyshown in Fig. 12.

The phase operator, (58), is monodimensional and can beupdated at each range bin,tb, in the range Doppler domain.However, it is safe to keep the full expression, with noapproximations:

Hφ(ω, k; tb) = exp

(jω0tb

(1−

√1− 4h2 cos2 θ

t2bc2

))(60)

The limitation embedded in (60) is due to the variation ofthe smile delay within one range resolution cell, that shouldgive a negligible phase error. If we approximate the operator(58):

Hφ ' exp(−j

ω0

c

2h2

tbccos2 θ0

)(61)

and we impose that the phase error to be withinπ in oneresolution cell,1/Br (Br being the range bandwidth), weget:∣∣∣∣∂ arg(Hφ)

∂t

∣∣∣∣ 1Br

< π ⇒∣∣∣∣2ω0

c2

h2

t2bcos2 θ0

∣∣∣∣ < πBr (62)

2h

ctb<

√Br

f0

As an example, in a microwave SAR with 20 MHz bandwidthand operating inC band (5.3 GHz), the ratio betweenthe transmitter-receiver separation and the bistatic closestapproach should be lower than 0.06, corresponding to aclosest approach larger than 300 km with a bistatic distance2h = 20 km. Notice that the limit becomes less stringent asthe bandwidth increases.

11

Finally, notice that as the phase operator (58) is time-varying. It cannot be simply interchanged with the pre-migration (57): if we implement first this second one then weshould account for it in the parametertb when implementing(58).

A. Focusing

The bistatic raw data, converted into an equivalent monos-tatic field by the preprocessing just described, can be focusedby any standard technique already developed for monostaticSAR, but some tuning of the processor parameters is re-quired.

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 con-version), therefore the central wavenumber, or the DopplerCentroid, should be computed from the bistatic geometry ofthe acquisition system. We need to derive with respect ofazimuth time the Doppler phase history described by the flat-top hyperbola:

φ(τ) =ω0

c

(√ζ2 + (vsatτ − h)2 +

√ζ2 + (vsatτ + h)2

)(63)

whereζ is the monostatic closest approach (e.g. with respectto the center of the bistatic system), andvτ corresponds tothe azimuth,y, (τ being the slow time), as shown in thegeometry of Fig. 3. The bistatic fm-rate is:

fi(τ) =vsat

λ

((vsatτ − h)√

ζ2 + (vsatτ − h)2+

(vsatτ + h)√ζ2 + (vsatτ + h)2

)=

vsat

λ(sin θr + sin θs) (64)

where the anglesθr and θs are defined in Fig. 11. Theactual Doppler Centroid should be computed by (64) incorrespondence of the azimuth timeτ that marks the center ofthe beam-width in the acquisition geometry (the illuminatedportion of the flat top hyperbola).

As a second aspect, we must remark that, although thesmile performs a (non-uniform) compression of the receivedimage, the time reference of the output is not to be changed.This implies that the focusing processor should computethe location of the focused targets by using the bistatictime labeling, and not the one that would come out fromthe monostatic-converted data set. In other words, for anunsquinted geometry, an hyperbola whose vertex is at rangebin l corresponds to a target with closest approach:

ζ =12

√(tswst +

l

fs

)2

c2 + h2 (65)

(tswst being the sampling window start time), and notζ =c (tswst + l/fs) /2 as one would expect in a monostaticsystem. 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,h,this shift should be made range variant, that would be quiteeasy accomplished in a range-Doppler processor, whereas itwould require a modification of the Stolt interpolation or ofthe chirp scaling function in a wavenumber domain processor(see [17] [18]).

B. Results from simulations

A preliminary validation of the technique here presentedhas been made by numerical simulation. A C band bistaticSAR has been designed by assuming the usual geometry inFig. 3, and with the parameters in Tab. XII-B. The choiceof parameters has been made in order to check a worst-case condition, where range migration is considerable and thebistatic system is quite different from a monostatic one. Thesimulated data set has been preprocessed as described before,and focused by a standard wavenumber-domain monostaticprocessor (with a careful Stolt interpolation).

Parameter ValuePRF 25 kHzSampling Frequency 50 MHzSensors Velocity 7.2 km/sCarrier Frequency 5.3 GHzSensor Altitude 20 kmh 3 kmSwath Depth 1.5 kmSynthetic Aperture 2.5 km

The result achieved after focusing a set of point targetsaligned along azimuth at squint angleθ = 0 (hence,y = 0),and equally displaced in slant range, is shown in Fig. 13 Thetarget are well focused, getting the resolution compatible withthe 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 achievedby processing the data set as monostatic, e.g. by assumingthe hyperbola that “approaches” the flat-top one, is in Fig.14.

As a further examples, the same bistatic system has beenassumed, but with a squint angleθ = 10o. The result isreported in Figs. 15 and 16: note that, without the properprocessing, targets are defocused (less than for zero squintas the bistatic effect is less marked), and also misplaced bothin range and in azimuth.

XIII. A CKNOWLEDGMENTS

The authors whish to thank Dr. Sergey Fomel, who ac-knowledged himself as a revisor, for his accurate revision,for his help in fixing some typos and for his suggestions andliterature reference.

12

Fig. 13. Focusing a set of point targets using the proposed pre-processing.Amplitude of the focused image in dB. The targets are well focused and inthe correct location (marked as a star in the image). Squint angleθ = 0.

Fig. 14. Focusing a set of point targets without using the proposed pre-processing, and just tuning the monostatic closest approach, same datasetas the one assumed in Fig. 13. The targets are defocused and not properlylocated.

Fig. 15. Focusing a set of point targets by exploiting the proposed pre-processing. The targets are well focused and in the correct location (markedas a star in the image). Squint angleθ = 100.

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

13

XIV. C ONCLUSIONS

In the paper we have shown that the framework devel-oped in geophysics since the early eighties for modelingand processing bistatic (and multistatic) systems can beported with success to the case of microwave SARs. Thisframework establishes a relation between a bistatic systemand an equivalent monostatic one centered in the midpointof the two sensors. The conversion is performed with noapproximation by a time-varying pre-processor, that acts asa sort of motion compensation (actually it can be used alsofor motion compensation). Preliminary simulations confirmsthe feasibility of the described technique even in worst caseconditions, of squinted systems with large migration andsignificant bistatic apertures. Hints are proposed to extendthe idea to non-stationary systems (not the case of futurespaceborne constellations).

We think that the principal advantage introduced by theproposed bistatic-to-monostatic conversion is not the re-useof the existing the SAR processors, but rather the extensionto the bistatic SAR of the zero-Doppler coordinate systems,with the ensuing simplifications in the design of futurespaceborne systems and in the geolocation.

APPENDIX

DMO DERIVATION BY STATIONARY PHASE

In this appendix we derive the decomposition of theforward bistatic model into the equivalent monostatic oneand the DMO operator by a formal point of view. Thederivation here shown is a porting to the SAR communityof the work of [30], that is in turn based on Hale [31]. Thefinal result generalizes that of Soumekh ([8], eq. (19),(40))and it does not involve series expansion, nor small squintapproximations.

Let us start from the impulse response of the bistatic SARacquisition, we simply extend the same model shown in [18]to the bistatic case. Let us replace they axis in the modelspace, with thex axis, that has a just a different origin. Inthis new reference, we will define the source and receiverlocation asxs, xr respectively:

y =xs + xr

2h =

xs − xr

2(66)

The impulse response of a target located in(x = 0, ζ) is theflat-top hyperbola:

hb(t, x) = δ

(t− Rs + Rr

c

)exp

(−jω0

Rs + Rr

c

)(67)

where the suffixb stands for ”bistatic”, andRs andRr are thetarget-to-source and target-to-receiver distance, respectively:

Rs =√

ζ2 + (x− xs)2

Rr =√

ζ2 + (x− xr)2

The summationRs + Rr leads to the DSR term.Let first transform such impulse response along time,t

htb(ω, x) = exp

(−j

(ω + ω0

c

)(Rs + Rr)

)(68)

= exp(−jω′

c

√ζ2 + (x− xs)

2)·

exp(−jω′

c

√ζ2 + (x− xr)

2)

where we have introduced the definitionω′ = ω + ω0, andthe superscript“t” to indicate the FT respect to that variable.

We then express each of the two factors in (68) asa superposition of plane waves. We exploit the followinginverse FT:

exp(−jω′

c

√ζ2 + ξ2) =∫

exp

−jζ

√(ω′

c

)2

− k2

exp(jkξ)dk (69)

that is quite known in SAR community, as it is exploited toderive the transfer function for monostatic SAR (see [8],[18]for a review). Notice that in (68) we have ignored complexconstants and we have neglected slow varying amplitudeterms, a quite reasonable assumption in SAR systems.

Combining (67) and (68) we get the decomposition of thereceived field into the wavenumber domains of the sourceand the receiver:

htb(ω

′, x) =∫∫

exp

−jζ

√(ω′

c

)2 − k2s+√(

ω′

c

)2 − k2r

· exp(j (ksxs + krxr))dksdkr (70)

Let then apply the change of coordinates (66) and, as aconsequence, introducing the corresponding wavenumbers (inthe Appendix we useky instead ofk, for clarity):

kg = ky/2 + kh/2ks = ky/2− kh/2

The received field (70) is now expressed in the (ky, kh)wavenumber domain an furthermore transformedy ⇔ ky:

ht,yb (ω′, y; ζ) =

∫exp

−jζ

√(ω′

c

)2 − (ky+kh)2

4

+√(

ω′

c

)2 − (ky−kh)2

4

(71)

· exp(jkhh)dkh

We follow now the same approach underlying the DMOderivation: instead than trying to invert the forward modelof the bistatic SAR (71) (that would be quite complicatedto derive in a closed form, and computationally expensive),

14

we rather convert that model into an equivalent monostaticone. The monostatic SAR acquisition is simply derived bysubstituting2r0 = (Rs + Rr) in (68) and following thesubsequent plane wave expansion (70), that would keep onlyone term:

hτ,ym (Ω′, ky) = exp

−j2r0

√(Ω′

c

)2

−k2

y

4

(72)

τ andΩ being the monostatic time and its companion angularvelocity.

It is possible to show that, under the change of variable:

ω′ = Ω′

√√√√1 +k2

h(Ω′

c/2

)2

− k2y

(73)

the following relation holds:

ζ

√(ω′

c

)2 − (ky+kh)2

4

+ζ

√(ω′

c

)2 − (ky−kh)2

4

= r0

√(Ω′

c/2

)2

− k2y (74)

(it can be proven just by squaring both members and thenapplying (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 forh = 0:

hτ,ym (Ω′, ky, ζ) =

∫ht,y

b (ω′ (Ω′, ky, kh) ; ζ) exp(jkhh)dkh

∣∣∣∣h=0

=∫

ht,y,hb (ω′ (Ω′, ky, kh) ; ζ) dkh (75)

Notice that the term on the left is the monostatic field(72), that shares the same center of the bistatic(h = 0) ,and the same wavenumber domain. However, in the changeof variable Ω → ω implied in (75) we have ignored theJacobean:

J =∣∣∣∣dΩ′

dω′

∣∣∣∣ (76)

=

√√√√1 +k2

h(Ω′

c/2

)2

− k2y

1−k2

hk2y((

Ω′

c/2

)2

− k2y

)2

that would result in a amplitude term that changes slowlywith time for the bistatic SAR of interest.

The derivation of (75) is thus pretty general, and it is notlimited to small squint angles or small apertures. It involvesan integration in thekh domain of the 3D transformed field,a change of coordinatesΩ → ω, and an inverse FTkh → h.Similarly to the Stolt interpolation [16], the kernel impliedby these steps is range variant. Let us evaluate the kerneldirectly in the time domain: we compute its (time-varying)

impulse response by assuming a bistatic field centered onsomet = tb

db(t, y) = δ (t− tb) δ(x) (77)

whose FT is clearly

dt,yb (ω′, ky) = exp (−jω′tb) (78)

We then apply the change of variable and the integration onkh involved in (75) getting the expression of the kernel inthe (Ω′, ky) domain:

dt,ym (Ω′, ky) =

∫exp(−jω′(Ω′, ky, kh)tb) exp(jkhh)dkh

(79)Eventually, we can get an explicit form for this kernel byexploiting the stationary phase to compute the integral. Letus write the phase term:

Φ(kh) = −Ω′

√√√√1 +k2

h(Ω′

c/2

)2

− k2y

tb + khh (80)

The stationary phase approximation of (79) is then:

dt,ym (Ω′, ky) ' D0 exp

(jφ(kh)

)(81)

D0 =

√2π

Ω′..

φ(kh)exp

(jπ

4sign

..

φ(kh))

to be computed in the stationary pointskh, solutions of:

dΦdkh

= 0 (82)

These solutions can be expressed in the following form:

kh = ±4h(Ω′2 − c2k2

y

)c2

√Ω′2(t2b −

(2hc

)2)+ h2k2

y

(83)

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

Φ(kh) = −Ω′

√√√√√1 +kh(

Ω′

c/2

)2

− k2y

tb + khh (84)

= −Ω′

√t2b −

4h2

c2+

h2k2y

Ω′2

Henceforth, if we ignore constant term, the time-varyingkernel is

dt,ym (Ω′, ky) ∝ exp

(−jΩ′

√t2b −

4h2

c2+

h2k2y

Ω′2

)(85)

If we approximate the last term into the square root as

h2k2y

Ω′2 '(

h∆f

vsatf0

)2

(86)

15

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

tm =

√t2b −

4h2

c2+(

h∆f

vsatf0

)2

(87)

that is the same result as in (35).

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