Microlocal aspects of bistatic synthetic aperture radar imaging
aa r X i v : . [ m a t h . A P ] A ug MICROLOCAL ASPECTS OF BISTATIC SYNTHETICAPERTURE RADAR IMAGING
Abstract.
In this article, we analyze the microlocal properties of the lin-earized forward scattering operator F and the reconstruction operator F ∗ F appearing in bistatic synthetic aperture radar imaging. In our model, theradar source and detector travel along a line a fixed distance apart. We showthat F is a Fourier integral operator, and we give the mapping properties of theprojections from the canonical relation of F , showing that the right projectionis a blow-down and the left projection is a fold. We then show that F ∗ F is asingular FIO belonging to the class I , . Venky P. Krishnan* and Eric Todd Quinto*
Department of Mathematics, Tufts University503, Boston Avenue, Medford, MA 02155, USA Introduction
In synthetic aperture radar (SAR) imaging, a region of interest on the surface ofthe earth is illuminated by electromagnetic waves from a moving airborne platform.The goal is to reconstruct an image of the region based on the measurement ofscattered waves. For an in-depth treatment of SAR imaging, we refer the readerto [3, 2]. SAR imaging is similar to other imaging problems such as Sonar whereacoustic waves are used to reconstruct the shape of objects on the ocean floor[1, 5, 19].In monostatic SAR, the source and the receiver are located on the same movingairborne platform. In bistatic SAR, the source and the receiver are on independentlymoving airborne platforms. There are several advantages to considering such dataacquisition geometries. The receivers, compared to the transmitters, are passiveand hence are more difficult to detect. Hence, by separating their locations, thereceivers alone can be in an unsafe environment, while the transmitters are in a safeenvironment. Furthermore, bistatic SAR systems are more resistant to electroniccountermeasures such as target shaping to reduce scattering in the direction ofincident waves [17].In this paper, we consider a bistatic SAR system where the antennas have poordirectivity and hence the beams do not focus on targets on the ground. We assumethat the transmitter and receiver traverse a 1-dimensional curve and the back-scattered data is measured at each point on this curve for a certain period of time.As in the monostatic SAR case [21], with a weak scattering assumption, the linearscattering operator that relates the unknown function that models the object on theground to the data at the receiver (see [25]) is a Fourier integral operator (FIO) [16,18, 4, 24]. Now if F is an FIO, the canonical relation Λ F associated to F tells us how Mathematics Subject Classification.
Primary: 58F15, 58F17; Secondary: 53C35.
Key words and phrases.
SAR Imaging, Radar, Fourier Integral Operators, Microlocal Analysis,Scattering.* Both authors were partially supported by NSF grant DMS 0908015. the singularities of the object are propagated to the data. The canonical relationΛ F ∗ of the L adjoint F ∗ of F gives us information as to how the singularities inthe data are propagated back to the reconstructed object. The microlocal analysisof singularities of the object is then done by analyzing the composition Λ F ∗ ◦ Λ F .Such an analysis for monostatic SAR has been done by several authors [22, 6,7] and is fairly well understood. In their work [22], Nolan and Cheney showedthat the composition of the linearized scattering operator with its L adjoint isa singular pseudodifferential operator (ΨDO) and it belongs to class of Fourierintegral operators with two cleanly intersecting Lagrangians. Felea in her works[6, 7] further analyzed the properties of the composition of these operators.In this paper, we do a similar analysis for the bistatic SAR imaging problem.Given the complications that arise in treating arbitrary transmitter and receivertrajectories, in this paper, we focus on the case where the transmitter and receiverare at the same height above the ground, traverse the same linear trajectory atthe same constant speed and spaced apart from each other by a constant distance.Furthermore, we assume that the object to be imaged is on the ground, which forsimplicity, we will assume is flat. Since the measured data is two-dimensional, it isreasonable to expect that we can reconstruct a two-dimensional object.The outline of the paper is as follows. Section 2 focuses on the preliminaries.Here we give the linearized scattering model for bistatic SAR and definitions ofsingularities and distributions that belong to I p,l classes and important results ondistributions belonging to this class that are required in this paper. In Section 3,we undertake a detailed study of the canonical relation associated this FIO. Thisis the content of Theorem 3.5. Then we study the reconstruction operator, thatis, the composition of the bistatic scattering operator with its adjoint, and show inTheorem 4.3 that this operator belongs to the class I , . Our proof of 4.3 followsthe ideas of [6, Theorem 1.6]. Several identities required to prove Theorem 4.3 areprovided in the Appendix. 2. Preliminaries
The bistatic linearized scattering model.
We assume that a bistatic SARsystem is involved in imaging a scene. Let γ T ( s ) and γ R ( s ) for s ∈ ( s , s ) be thetrajectories of the transmitter and receiver respectively. The transmitter transmitselectromagnetic waves that scatter off the target, which are then measured at thereceiver. We are interested in obtaining a linearized model for this scattered signal.The propagation of electromagnetic waves can be described by the scalar waveequation: (cid:18) ∆ − c ∂ t (cid:19) E ( x, t ) = P ( t ) δ ( x − γ T ( s )) , where c is the speed of electromagnetic waves in the medium, E ( x, t ) is the electricfield and P ( t ) is the transmit waveform sent to the transmitter antenna located atposition γ T ( s ). The wave speed c is spatially varying due to inhomogeneities presentin the medium. We assume that the background in which the electromagnetic wavespropagate is free space. Therefore c can be expressed as:1 c ( x ) = 1 c + e V ( x ) , ISTATIC SAR IMAGING 3 where the constant c is the speed of light in free space and e V ( x ) is the perturba-tion due to deviation from the background, which we would like to recover frombackscattered waves.Since the incident electromagnetic waves in typical radar frequencies attenuaterapidly as they penetrate the ground, we assume that e V ( x ) varies only on a 2-dimensional surface. Therefore, we represent e V as a function of the form e V ( x ) = V ( x ) δ ( x )where we assume for simplicity that the earth’s surface is flat, represented by the x = ( x , x ) plane.The background Green’s function g is then given by the solution to the followingequation: (cid:18) ∆ − c ∂ t (cid:19) g ( x, γ T ( s ) , t ) = δ ( x − γ T ( s )) δ ( t ) . We can explicitly write g as g ( x, γ T ( s ) , t ) = δ ( t − | x − γ T ( s ) | /c )4 π | x − γ T ( s ) | . Now the incident field E in due to the source s ( x, t ) = P ( t ) δ ( x − γ T ( s )) is E in ( x, t ) = Z g ( x, y, t − τ ) s ( y, τ )d y d τ = P ( t − | x − γ T ( s ) | /c )4 π | x − γ T ( s ) | . Let E denote the total field of the medium, E = E in + E sc . Then the scatteredfield can be written using the Lippman-Schwinger equation:(1) E sc ( z, t ) = Z g ( z, x, t − τ ) ∂ t E ( x, τ ) T ( x )d x d τ. We linearize (1) by the first Born approximation and write the linearized scatteredwave-field at receiver location γ R ( s ): E sclin ( γ R ( s ) , t ) = Z g ( γ R ( s ) , x, t − τ ) ∂ t E in ( x, τ ) V ( x )d x d τ = Z δ ( t − τ − | x − γ R ( s ) | /c )4 π | x − γ R ( s ) | (cid:18) e − i ω ( τ −| x − γ T ( s ) | /c ) ω p ( ω )4 π | x − γ T ( s ) | (cid:19) (2) × V ( x )d ω d x d τ, where p is the Fourier transform of P .Now, integrating (2) with respect to τ , a linearized model for the scattered signalis as follows:(3) d ( s, t ) := E sclin ( γ R ( s ) , t ) = Z e − i ω ( t − c R ( s,x )) A ( s, x, ω ) V ( x )d x d ω, where(4) R ( s, x ) = | γ T ( s ) − x | + | x − γ R ( s ) | and(5) A ( s, x, ω ) = p ( ω )((4 π ) | γ T ( s ) − x || γ R ( s ) − x | ) − . This function includes terms that take into account the transmitter and receiverantenna beam patterns, the transmitted waveform and geometric spreading factors.
VENKY P. KRISHNAN AND ERIC TODD QUINTO
We will show in Section 3 in one important case, that the transform F thatmaps V to (3) is a Fourier integral operator associated to a canonical relation Λ(Proposition 3.1), and we will prove the mapping properties of Λ (Propositions 3.2and 3.4). These mapping properties tell what F does to singularities. We nowdefine the mapping properties we need.2.2. Singularities and I p,l classes. Here we give the definitions of the singular-ities associated with our operator F and its canonical relation (11), and a class ofdistributions required for the analysis of the composition F with its L adjoint. Definition 2.1. [11] Let M and N be manifolds of dimension n and let f : N → M be C ∞ .(1) f is a Whitney fold if near each m ∈ M , f is either a local diffeomorphismor f drops rank simply by one at m so that L = { m ∈ M : rank df = n − } is a smooth hypersurface at m and ker df ( m ) T m L .(2) f is a blow-down along a smooth hypersurface L ⊂ M if f is a local dif-feomorphism away from L and f drops rank simply by one at L , wherethe Hessian of f is equal to zero and ker df ⊂ T ( L ), so that f (cid:12)(cid:12) L has one-dimensional fibers.We now define I p,l classes. They were first introduced by Melrose and Uhlmann,[20] Guillemin and Uhlmann [15] and Greenleaf and Uhlmann [10] and they wereused in the context of radar imaging in [22, 6, 7]. Definition 2.2.
Two submanifolds M and N intersect cleanly if M ∩ N is a smoothsubmanifold and T ( M ∩ N ) = T M ∩ T N .Consider two spaces X and Y and let Λ and Λ and ˜Λ and ˜Λ be Lagrangiansubmanifolds of the product space T ∗ X × T ∗ Y . If they intersect cleanly (˜Λ , ˜Λ )and (Λ , Λ ) are equivalent in the sense that there is, microlocally, a canonicaltransformation χ which maps (Λ , Λ ) into (˜Λ , ˜Λ ). This leads us to the followingmodel case. Example 2.3.
Let ˜Λ = ∆ T ∗ R n = { ( x, ξ ; x, ξ ) | x ∈ R n , ξ ∈ R n \ } be the diagonalin T ∗ R n × T ∗ R n and let ˜Λ = { ( x ′ , x n , ξ ′ , x ′ , y n , ξ ′ , | x ′ ∈ R n − , ξ ′ ∈ R n − \ } .Then, ˜Λ intersects ˜Λ cleanly in codimension . Now we define the class of product-type symbols S p,l ( m, n, k ). Definition 2.4. S p,l ( m, n, k ) is the set of all functions a ( z, ξ, σ ) ∈ C ∞ ( R m × R n × R k ) such that for every K ⊂ R m and every α ∈ Z n + , β ∈ Z n + , γ ∈ Z k + there is c K,α,β such that | ∂ αz ∂ βξ ∂ γσ a ( z, ξ, σ ) | ≤ c K,α,β (1 + | ξ | ) p −| β | (1 + | σ | ) l −| γ | , ∀ ( z, ξ, τ ) ∈ K × R n × R k . Since any two sets of cleanly intersecting Lagrangians are equivalent, we firstdefine I p,l classes for the case in Example 2.3. Definition 2.5. [15] Let I p,l (˜Λ , ˜Λ ) be the set of all distributions u such that u = u + u with u ∈ C ∞ and u ( x, y ) = Z e i (( x ′ − y ′ − s ) · ξ ′ +( x n − y n − s ) · ξ n + s · σ ) a ( x, y, s ; ξ, σ ) dξdσds with a ∈ S p ′ ,l ′ where p ′ = p − n + and l ′ = l − . ISTATIC SAR IMAGING 5
This allows us to define the I p,l (Λ , Λ ) class for any two cleanly intersectingLagrangians in codimension 1 using the microlocal equivalence with the case inExample 2.3. Definition 2.6. [15] Let I p,l (Λ , Λ ) be the set of all distributions u such that u = u + u + P v i where u ∈ I p + l (Λ \ Λ ), u ∈ I p (Λ \ Λ ), the sum P v i is locally finite and v i = F w i where F is a zero order FIO associated to χ − , thecanonical transformation from above, and w i ∈ I p,l (˜Λ , ˜Λ ).This class of distributions is invariant under FIOs associated to canonical trans-formations which map the pair (Λ , Λ ) to itself. By definition, F ∈ I p,l (Λ , Λ ) if itsSchwartz kernel belongs to I p,l (Λ , Λ ). If F ∈ I p,l (Λ , Λ ) then F ∈ I p + l (Λ \ Λ )and F ∈ I p (Λ \ Λ ) [15]. Here by F ∈ I p + l (Λ \ Λ ), we mean that the Schwartzkernel of F belongs to I p + l (Λ \ Λ ) microlocally away from Λ .3. Transmitter and receiver in a linear trajectory
Henceforth, let us assume that the trajectory of the transmitter is γ T : ( s , s ) → R , γ T ( s ) = ( s + α, , h )and the trajectory of the receiver is γ R ( s ) : ( s , s ) → R , γ R ( s ) = ( s − α, , h ) . Here α > h > s ∈ ( s , s ) and t ∈ ( t , t ) is(6) d ( s, t ) = Z e − i ω (cid:16) t − c ( | x − γ T ( s ) | + | x − γ R ( s ) | ) (cid:17) A ( s, x, ω ) V ( x )d x d ω. We multiply d ( s, t ) by a smooth (infinitely differentiable) function f ( s, t ) supportedin a compact subset of ( s , s ) × ( t , t ). This compensates for the discontinuities inthe measurements at the end points of the rectangle ( s , s ) × ( t , t ). For simplicity,let us denote the function f · d as d again. We then have(7) d ( s, t ) = Z e − i ω ( t − c R ( s,x )) A ( s, t, x, ω ) V ( x )d x d ω, where now A ( s, t, x, ω ) = f ( s, t ) A ( s, x, ω ).Our method cannot image the point on the object that is “directly underneath”the transmitter and the receiver. This is, if the transmitter and receiver are atlocations ( s + α, , h ) and ( s − α, , h ), then we cannot image the point ( s, , d in Equation (7) by multiplying by another smooth function g ( s, t ) such that g ≡ ( s, √ α + h c ! : s < s < s ) . For simplicity, again denote g · d as d and g · A as A . Consider,(8) F V ( s, t ) := d ( s, t ) = Z e − i ω ( t − c ( | x − γ T ( s ) | + | x − γ R ( s ) | )) A ( s, t, x, ω ) V ( x )d x d ω. For simplicity, let us denote the ( s, t ) space as Y .We assume that the amplitude function A satisfies the following estimate: Forevery compact K ∈ Y × X and for every non-negative integer α and for every2-indexes β = ( β , β ) and γ , there is a constant C such that(9) | ∂ αω ∂ β s ∂ β t ∂ γx A ( s, t, x, ω ) | ≤ C (1 + | ω | ) − α . VENKY P. KRISHNAN AND ERIC TODD QUINTO
The phase function of the operator F ,(10) ψ ( s, t, x, ω ) = − ω (cid:18) t − c ( | x − γ T ( s ) | + | x − γ R ( s ) | ) (cid:19) is homogeneous of degree 1 in ω .We now analyze some properties of the canonical relation of the operator F . Proposition 3.1. F is a Fourier integral operator of order 3/2 associated withcanonical relation Λ = ( (cid:18) s, t, − ωc (cid:18) x − s − α | x − γ T ( s ) | + x − s + α | x − γ R ( s ) (cid:19) , − ω (cid:19) ;(11) x , x , − ωc (cid:18) x − s − α | x − γ T ( s ) | + x − s − α | x − γ R ( s ) | (cid:19) , − ωc (cid:18) x | x − γ T ( s ) | + x | x − γ R ( s ) | (cid:19)! : c t = q ( x − s − α ) + x + h + q ( x − s + α ) + x + h , ω = 0 ) . Furthermore ( x , x , s, ω ) is a global parameterization for Λ .Proof. This is a straightforward application of the theory of FIO. Since ψ in (10) isa nondegenerate phase function with ∂ x ψ and ∂ s,t ψ nowhere zero and the amplitude A in (8) is of order 2, F is an FIO [16]. Since the amplitude is of order 2, the orderof the FIO is 3/2 by [16, Definition 3.2.2]. By definition [16, Equation (3.1.2)]Λ = { ( s, t, ∂ s,t ψ ( x, s, t, ω )) , ( x, − ∂ x ψ ( x, s, t )) : ∂ ω ψ ( x, s, t, ω ) = 0 } . A calculation using this definition establishes (11). Finally, it is easy to see that( x , x , s, ω ) is a global parameterization of Λ. (cid:3) In order to understand the microlocal mapping properties of F and F ∗ F , weconsider the projections π L : T ∗ Y × T ∗ X → T ∗ Y and π R : T ∗ Y × T ∗ X → T ∗ X . Proposition 3.2.
The projection π L restricted to Λ has a fold singularity on Σ := { ( x, , s, ω ) : ω = 0 } .Proof. The projection π L is given by(12) π L ( x , x , s, ω ) == s, c ( | x − γ T ( s ) | + | x − γ R ( s ) | ) , − ωc (cid:18) x − s − α | x − γ T ( s ) | + x − s + α | x − γ R ( s ) (cid:19) , − ω ! We have( π L ) ∗ = c (cid:16) x − s − α | x − γ T ( s ) | + x − s + α | x − γ R ( s ) | (cid:17) c (cid:16) x | x − γ T ( s ) | + x | x − γ R ( s ) | (cid:17) ∗ − ωc (cid:16) x + h | x − γ T ( s ) | + x + h | x − γ R ( s ) | (cid:17) ωc (cid:16) ( x − s − α ) x | x − γ T ( s ) | + ( x − s + α ) x | x − γ R ( s ) | (cid:17) ∗ ∗ − Thendet( π L ) ∗ = ωc x (cid:18) | x − γ T ( s ) | + 1 | x − γ R ( s ) | (cid:19) (cid:18) x − s ) + x + h − α | x − γ T ( s ) || x − γ R ( s ) | (cid:19) . ISTATIC SAR IMAGING 7
The proposition then follows as a consequence of the following lemma.
Lemma 3.3.
The term x − s ) + x + h − α | x − γ T ( s ) || x − γ R ( s ) | . is positive for all x ∈ R , s ∈ R and h and α positive.Proof. This is a straightforward calculation that is made simpler if one lets S = x − s , T = p x + h then puts the term over a common denominator. Finally, oneshows that the numerator is positive by isolating the square roots (absolute values),squaring, and simplifying to infer that, since 4 T α >
0, the lemma is true. (cid:3)
Now returning to the proof of the Proposition 3.2, we have that det( π L ) ∗ = 0 ifand only if x = 0. Hence det( π L ) ∗ vanishes on the set Σ and Lemma 3.3 againshows that d(det( π L ) ∗ ) on Σ is non-vanishing. This implies that π L drops ranksimply by one on Σ.Now it remains to show that T Σ ∩ Kernel( π L ) ∗ = { } . But this follows from thefact that Kernel( π L ) ∗ = span( ∂∂x ), but T Σ = span( ∂∂x , ∂∂s , ∂∂ω ). This concludesthe proof of the proposition. (cid:3) Proposition 3.4.
Consider the projection π R : T ∗ Y × T ∗ X → T ∗ X . The restric-tion of the projection to Λ has a blowdown singularity on Σ .Proof. We have(13) π R ( x , x , s, ω ) == x , x , − ωc (cid:18) x − s − α | x − γ T ( s ) | + x − s − α | x − γ R ( s ) | (cid:19) , − ωc (cid:18) x | x − γ T ( s ) | + x | x − γ R ( s ) | (cid:19) ! . Now( π R ) ∗ = ∗ ∗ − ωc (cid:16) x + h | x − γ T ( s ) | + x + h | x − γ R ( s ) | (cid:17) − c (cid:16) x − s − α | x − γ T ( s ) | + x − s + α | x − γ R ( s ) | (cid:17) ∗ ∗ ωc (cid:16) ( x − s − α ) x | x − γ T ( s ) | + ( x − s + α ) x | x − γ R ( s ) | (cid:17) − c (cid:16) x | x − γ T ( s ) | + x | x − γ R ( s ) | (cid:17) From this we see that Kernel( π R ) ∗ ⊂ T Σ. Since det(( π R ) ∗ ) = det(( π L ) ∗ ), π R dropsrank simply by one along Σ. Therefore the projection π R has a blowdown singularityalong Σ. (cid:3) We summarize what we have proved in this section by the following theorem:
Theorem 3.5.
The operator F defined in (8) is a Fourier integral operator oforder / . The canonical relation Λ associated to F defined in (11) satisfies thefollowing: The projections π L and π R defined in (12) and (13) are a fold andblowdown respectively. VENKY P. KRISHNAN AND ERIC TODD QUINTO Image Reconstruction
Next, we study the composition of F with F ∗ . This composition is given asfollows: F ∗ F T ( x ) = Z e i (cid:16) ω ( t − c ( | x − γ R ( s ) | + | x − γ R ( s ) | )) − e ω ( t − c ( | y − γ T ( s ) | + | y − γ R ( s ) | )) (cid:17) × A ( x, s, t, ω ) A ( y, s, t, e ω )d s d t d ω d e ω d y After an application of the method of stationary phase, we can write the kernel ofthe operator F ∗ F as K ( x, y ) = Z e i ωc ( | y − γ T ( s ) | + | y − γ R ( s ) |− ( | x − γ T ( s ) | + | x − γ R ( s ) | )) e A ( x, y, s, ω )d s d ω. Therefore the phase function of the kernel K ( x, y ) is(14) φ ( x, y, s, ω ) = ωc ( | y − γ T ( s ) | + | y − γ R ( s ) | − ( | x − γ T ( s ) | + | x − γ R ( s ) | )) . Proposition 4.1.
W F ( K ) ′ ⊂ ∆ ∪ Λ where ∆ := { ( x , x , ξ , ξ ; x , x , ξ , ξ ) } and Λ := { ( x , x , ξ , ξ ; x , − x , ξ , − ξ ) } .Here for a point x = ( x , x ) , the covectors ( ξ , ξ ) are non-zero multiples of thevector ( − ∂ x R ( s, x ) , − ∂ x R ( s, x )) , where R is defined in (4) .Proof. Using the H¨ormander-Sato Lemma, we have
W F ( K ) ′ ⊂⊂ ( x , x , − ωc (cid:18) x − s − α | x − γ T ( s ) | + x − s + α | x − γ R ( s ) | (cid:19) , − ωc (cid:18) x | x − γ T ( s ) | + x | x − γ R ( s ) | (cid:19)! ; y , y , − ωc (cid:18) y − s − α | y − γ T ( s ) | + y − s + α | y − γ R ( s ) | (cid:19) , − ωc (cid:18) y | y − γ T ( s ) | + y | y − γ R ( s ) | (cid:19)! : | x − γ T ( s ) | + | x − γ R ( s ) | = | y − γ T ( s ) | + | y − γ R ( s ) | ,x − s − α | x − γ T ( s ) | + x − s + α | x − γ R ( s ) | = y − s − α | y − γ T ( s ) | + y − s + α | y − γ R ( s ) | , ω = 0 ) . We now obtain a relation between ( x , x ) and ( y , y ). This is given by the followinglemma. Lemma 4.2.
For all s , the set of all ( x , x ) , ( y , y ) that satisfy | x − γ T ( s ) | + | x − γ R ( s ) | = | y − γ T ( s ) | + | y − γ R ( s ) | , (15) x − s − α | x − γ T ( s ) | + x − s + α | x − γ R ( s ) | = y − s − α | y − γ T ( s ) | + y − s + α | y − γ R ( s ) | . (16) necessarily satisfy the following relations: x = y and x = ± y .Proof. In order to show this, we will consider (15) and (16) as functions of R byreplacing h in these expressions with x − h . We then transform these expressionsusing the coordinates (17) and then set x = y = 0 to prove the lemma.Consider the following change of coordinates:(17) x = s + α cosh ρ cos θ y = s + α cosh ρ ′ cos θ ′ x = α sinh ρ sin θ cos ϕ y = α sinh ρ ′ sin θ ′ cos ϕ ′ x = h + α sinh ρ sin θ sin ϕ y = h + α sinh ρ ′ sin θ ′ sin ϕ ′ ISTATIC SAR IMAGING 9 where s , α > h > ρ ∈ [0 , ∞ ), θ ∈ [0 , π ] and ϕ ∈ [0 , π ). Thisa well-defined coordinate system except for ξ = 0 and θ = 0 , π .In the coordinate system (17), we have(18) | x − γ T ( s ) | = α (cosh ρ − cos θ ) , | x − γ R ( s ) | = α (cosh ρ + cos θ ) , x − s − α | x − γ T ( s ) = cosh ρ cos θ − ρ − cos θ , x − s + α | xγ R ( s ) | = cosh ρ cos θ +1cosh ρ +cos θ . The terms involving y are obtained similarly. Now (15) and (16) transform asfollows:2 cosh ρ = 2 cosh ρ ′ cosh ρ cos θ − ρ − cos θ + cosh ρ cos θ + 1cosh ρ + cos θ = cosh ρ ′ cos θ ′ − ρ ′ − cos θ ′ + cosh ρ ′ cos θ ′ + 1cosh ρ ′ + cos θ ′ . Using the first equality in the second equation, we havecos θ cosh ρ − cos θ = cos θ ′ cosh ρ − cos θ ′ . This gives cos θ = cos θ ′ . Therefore θ = 2 nπ ± θ ′ , which then gives sin θ = ± sin θ ′ .Therefore, in terms of ( x , x ) and ( y , y ), we have x = y and x = ± y . (cid:3) Now to finish the proof of the proposition, when x = y and x = y , there is con-tribution to W F ( K ) ′ contained in the diagonal set ∆ := { ( x , x , ξ , ξ ; x , x , ξ , ξ ) } and when x = y and x = − y , we have a contribution to W F ( K ) ′ contained inΛ, where Λ := { ( x , x , ξ , ξ ; x , − x , ξ , − ξ ) } . (cid:3) We use the following convention in the theorem below. The cotangent variablescorresponding to x and y are denoted as ξ and η respectively. Then note that ξ = ∂ x φ and η = − ∂ y φ .We now prove the following theorem: Theorem 4.3.
Let F be given by (8) . Then F ∗ F ∈ I , (∆ , Λ) .Proof. We follow the proof of [6, Theorem 1.6] closely to prove this result. Recallthat ∆ and Λ are defined by∆ = { x − y = x − y = ξ − η = ξ − η = 0 } , Λ = { x − y = x + y = ξ − η = ξ + η = 0 } . The ideal of functions that vanish on ∆ ∪ Λ is generated by e p = x − y , e p = x − y , e p = ξ − η , e p = ( x + y )( ξ − η ) , e p = ( x − y )( ξ + η ) , e p = ξ − η . Let p i = q i e p i , for 1 ≤ i ≤
6, where q , q are homogeneous of degree 1 in ( ξ, η ), q , q and q are homogeneous of degree 0 in ( ξ, η ) and q is homogeneous of degree − ξ, η ). Let P i be pseudodifferential operators with principal symbols p i for1 ≤ i ≤ F ∗ F ∈ I p,l (∆ , Λ), for some p, l , we have to show that P i K ∈ H s loc for some s , for 1 ≤ i ≤
6. By [4, Proposition 4.3.1], we have thefollowing (up to lower order terms): P i K ( x, y ) = Z e i ωc ( R ( s,y ) − R ( s,x )) e A ( x, y, s, ω ) p i ( x, y, − ∂ x R ( s, x ) , − ∂ y R ( s, y ))d s d ω We show in the Appendix that each e p i can be expressed in the following forms: e p = f ( x, y, s ) ω ∂ s φ + f ( x, y, s ) ∂ ω φ (19) e p = f ( x, y, s ) ω ∂ s φ + f ( x, y, s ) ∂ ω φ (20) e p = − ∂ s φ (21) e p = f ( x, y, s ) ∂ s φ + ωf ( x, y, s ) ∂ ω φ (22) e p = f ( x, y, s ) ∂ s φ + ωf ( x, y, s ) ∂ ω φ (23) e p = ωf ( x, y, s ) ∂ s φ + ω f ( x, y, s ) ∂ ω φ. (24)Here f ij for 1 ≤ i ≤ j = 1 , P K ( x, y ) = Z e i φ ( x,y,s,ω ) e A ( x, y, s, ω ) q (cid:18) f ( x, y, s ) ω ∂ s φ + f ( x, y, s ) ∂ ω φ (cid:19) d s d ω = Z ∂ s (cid:16) e i φ ( x,y,s,ω ) (cid:17) q i ω e A ( x, y, s, ω ) f ( x, y, s )d s d ω + Z ∂ ω (cid:16) e i φ ( x,y,s,ω ) (cid:17) q i e A ( x, y, s, ω ) f ( x, y, s )d s d ω By integration by parts= − ( Z e i φ ( x,y,s,ω ) ∂ s (cid:16) q i ω e A ( x, y, s, ω ) f ( x, y, s ) (cid:17) d s d ω + Z e i φ ( x,y,s,ω ) ∂ ω (cid:16) q i e A ( x, y, s, ω ) f ( x, y, s ) (cid:17) d s d ω ) . Note that q is homogeneous of degree 1 in ω , and e A is a symbol of order 4, henceeach amplitude term in the sum above is of order 4.Therefore by [4, Theorem 2.2.1], we have that P K ∈ H s loc .A similar argument works for each of the other five pseudodifferential operators.Hence by [10, Proposition 1.35], we have that F ∗ F ∈ I p,l (∆ , Λ). Because C is alocal canonical graph away from Σ, the transverse intersection calculus applies forthe composition F ∗ F away from Σ. Hence F ∗ F is of order 3 on ∆ \ Σ and Λ \ Σ.Since F ∗ F is of order p + l on ∆ \ Σ and is of order p on Λ \ Σ, we have that p = 3and l = 0. Therefore the theorem is proved. (cid:3) Acknowledgments
The first named author thanks Dave Isaacson, Margaret Cheney, Birsen Yazıcıand Art Weiss for discussions regarding this work while he was a post-doctoralfellow at RPI. Additionally, he thanks the Department of Mathematics at TuftsUniversity for providing a wonderful research environment. Both authors thankCliff Nolan, Raluca Felea, Allan Greenleaf, and Gunther Uhlmann for stimulatingdiscussions about mathematics related to this research.
ISTATIC SAR IMAGING 11
Appendix
A.Here we prove the identities (19) through (24) that are required in the proofof Theorem 4.3. For convenience, and without loss of generality, we will assume c = 1.In obtaining these identities, it is easier to work in the coordinate system definedin (17). We will work with the extension e φ of the phase function φ to R definedby e φ = ωc q ( y − s − α ) + y + ( y − h ) + p ( y − s + α ) + y y − h ) − (cid:18)q ( x − s − α ) + x + ( x − h ) + q ( x − s + α ) + x + ( x − h ) (cid:19) ! . Then, using the facts that(25) ∂ ω e φ | x = y =0 = ∂ ω φ and ∂ s e φ | x = y =0 = ∂ s φ, we set the third coordinate x = y = 0 to obtain the required identities.A.1. Expression for x − y . We now obtain an expression for x − y of the form(26) x − y = f ( x, y, s ) ω ∂ s φ + f ( x, y, s ) ∂ ω φ, where f and f are smooth functions.That is, denoting A = x − y , we would like to obtain an expression of theform (26) involving ∂ s φ and ∂ ω φ for(27) A = α (cosh ρ cos θ − cosh ρ ′ cos θ ′ ) . We have ∂ ω e φ = 2 α (cosh ρ ′ − cosh ρ )and ∂ s e φ = − ω (cid:18) cosh ρ ′ cos θ ′ − ρ ′ − cos θ ′ + cosh ρ ′ cos θ ′ + 1cosh ρ ′ + cos θ ′ (cid:19) − (cid:18) cosh ρ cos θ − ρ − cos θ + cosh ρ cos θ + 1cosh ρ + cos θ (cid:19) ! = 2 ω (cid:18) sinh ρ cos θ cosh ρ − cos θ − sinh ρ ′ cos θ ′ cosh ρ ′ − cos θ ′ (cid:19) (28)using cosh ρ cos θ − cos θ = sinh ρ cos θ . Now(29) cos θ ∂ ω e φ α (cosh ρ ′ cos θ ′ − cosh ρ cos θ ) + α cosh ρ ′ (cos θ − cos θ ′ ) . Then(30) A = − cos θ ∂ ω e φ + α cosh ρ ′ (cos θ − cos θ ′ ) . Adding and subtracting sinh ρ cos θ ′ cosh ρ − cos θ ′ inside (28), we have ∂ s e φ = 2 ω sinh ρ cos θ cosh ρ − cos θ − sinh ρ cos θ ′ cosh ρ − cos θ ′ | {z } I + sinh ρ cos θ ′ cosh ρ − cos θ ′ − sinh ρ ′ cos θ ′ cosh ρ ′ − cos θ ′ | {z } II ! . Simplifying I and II , we have, I = (cos θ − cos θ ′ )(sinh ρ )(cosh ρ + cos θ cos θ ′ )(cosh ρ − cos θ )(cosh ρ − cos θ ′ )and II = cos θ ′ sin θ ′ (cosh ρ − cosh ρ ′ )(cosh ρ + cosh ρ ′ )(cosh ρ ′ − cos θ ′ )(cosh ρ − cos θ ′ )= − cos θ ′ sin θ ′ (cosh ρ + cosh ρ ′ )(cosh ρ ′ − cos θ ′ )(cosh ρ − cos θ ′ ) · ∂ ω e φ α where we have used the fact sinh ρ − sinh ρ ′ = cosh ρ − cosh ρ ′ .Using these calculations, we seecos θ − cos θ ′ = (cid:18) (cosh ρ − cos θ )(cosh ρ − cos θ ′ )(cosh ρ − ρ + cos θ cos θ ′ ) (cid:19) (31) × ∂ s e φ ω + cos θ ′ sin θ ′ (cosh ρ + cosh ρ ′ )(cosh ρ ′ − cos θ ′ )(cosh ρ − cos θ ′ ) · ∂ ω e φ α ! . Now setting x = y = 0 and using (30) , we have, A = α cosh ρ ′ (cosh ρ − cos θ )(cosh ρ − cos θ ′ )(cosh ρ − ρ + cos θ cos θ ′ ) ∂ s φ ω − (cid:18) cos θ − cosh ρ ′ cos θ ′ sin θ ′ (cosh ρ + cosh ρ ′ )(cosh ρ − ρ + cos θ cos θ ′ ) · (cosh ρ − cos θ )(cosh ρ ′ − cos θ ′ ) (cid:19) ∂ ω φ. We can see that no denominator in this expression is zero for x = 0 (since cosh ρ > ≥ cos θ if x = 0) and so this expression for A is defined and smooth for allvalues of the coordinates.We can write A in the Cartesian coordinate system as follows. First, for sim-plicity, let X = | x − γ T ( s ) | = q ( x − s − α ) + x + h , (32) X = | x − γ R ( s ) | = q ( x − s + α ) + x + h , (33) Y = | y − γ T ( s ) | = q ( y − s − α ) + y + h , (34) Y = | y − γ R ( s ) | = q ( y − s + α ) + y + h . (35) ISTATIC SAR IMAGING 13
Then using these expressions and (18) we see that A = α (cid:0) Y + Y α (cid:1) (cid:0) X X α (cid:1) (cid:16) ( X + X ) α − y − s ) ( Y + Y ) (cid:17) ω (cid:16) ( X + X ) α − (cid:17) (cid:16) ( X + X ) α + x − s )( y − s )( X + X )( Y + Y ) (cid:17) ∂ s φ − x − s ) X + X − X X (cid:16) y − s ) Y + Y (cid:17) (cid:16) − y − s ) ( Y + Y ) (cid:17) (cid:0) X + X + Y + Y α (cid:1) Y Y (cid:16) ( X + X ) α − (cid:17) (cid:16) ( X + X ) α + x − s )( y − s )( X + X )( Y + Y ) (cid:17) ∂ ω φ. We see from this second expression for A that the coefficient functions are smoothin ( x, y, s ) since the expressions (32)-(35) are non-zero and smooth.A.2. Expression for x − y . Now we write x − y in the form(36) A := x − y = f ( x, y, s ) ω ∂ s φ + f ( x, y, s ) ∂ ω φ, where f and f are smooth functions. A in the coordinate system (17) is A : = α (sinh ρ sin θ cos ϕ − sinh ρ ′ sin θ ′ cos ϕ ′ )= α (sinh ρ sin θ − sinh ρ ′ sin θ ′ )(37) + α (sinh ρ ′ sin θ ′ sin ϕ ′ − sinh ρ sin θ sin ϕ ) . (38)For x = y = 0, (38) is 0. So it is enough to obtain an expression of the form (36)for (37), which we still denote by A .Using the following identities,sinh ρ = cosh ρ − θ = 1 − cos θ, we have, A = α (sinh ρ sin θ − sinh ρ ′ sin θ ′ )= α (cid:0) (cosh ρ − cosh ρ ′ ) + (cos θ − cos θ ′ ) − (cosh ρ cos θ − cosh ρ ′ cos θ ′ ) (cid:1) = − α ρ + cosh ρ ′ ) ∂ ω e φ + α (cos θ + cos θ ′ )(cos θ − cos θ ′ ) − α (cosh ρ cos θ + cosh ρ ′ cos θ ′ ) A = − α ρ + cosh ρ ′ ) ∂ ω e φ + α (cos θ + cos θ ′ )(cos θ − cos θ ′ ) − α (cosh ρ cos θ + cosh ρ ′ cos θ ′ )( − cos θ ∂ ω φ + α cosh ρ ′ (cos θ − cos θ ′ ))= α (cid:16) − (cosh ρ + cosh ρ ′ ) + (cosh ρ cos θ + cosh ρ ′ cos θ ′ ) cos θ (cid:17) ∂ ω e φ + α (cid:16) (cos θ + cos θ ′ ) − (cosh ρ cos θ + cosh ρ ′ cos θ ′ ) cosh ρ ′ (cid:17) (cos θ − cos θ ′ ) . Now we use the expression for cos θ − cos θ ′ in Equation (31) and set x = y = 0;this shows that x − y can be written in the form A = f ( x, y, s ) ω ∂ s φ + f ( x, y, s ) ∂ ω φ. A.3.
Expression for ξ − η . Now we consider ξ − η , where we recall that( ξ , ξ ) = ∂ x φ and ( η , η ) = − ∂ y φ are the cotangent variables corresponding to( x , x ) and ( y , y ) respectively.Then note that ξ − η is ∂ x φ + ∂ y φ . But this is the same as − ∂ s φ . Hence A := ξ − η = − ∂ s φ. A.4.
Expression for ( x − y )( ξ + η ) . We have (up to a negative sign)( x − y )( ξ + η ) = ω ( x − y ) (cid:18) x | x − γ T | + x | x − γ R | + y | y − γ T | + y | y − γ R | (cid:19) = 2 ω x cosh ρ cosh ρ − cos θ − y cosh ρ ′ cosh ρ ′ − cos θ ′ + x y cosh ρ ′ cosh ρ ′ − cos θ ′ − x y cosh ρ cosh ρ − cos θ ! = 2 ω x cosh ρ cosh ρ − cos θ − x cosh ρ ′ cosh ρ ′ − cos θ ′ + ( x − y ) cosh ρ ′ cosh ρ ′ − cos θ ′ + x y cosh ρ ′ cosh ρ ′ − cos θ ′ − x y cosh ρ cosh ρ − cos θ ! , Here we have added and subtracted x cosh ρ ′ cosh ρ ′ − cos θ ′ in the previous equation. Sim-plifying this we get,( x − y )( ξ + η ) =2 ω ( x − x y ) " (cosh ρ cosh ρ ′ + cos θ )(cosh ρ ′ − cosh ρ )(cosh ρ − cos θ )(cosh ρ ′ − cos θ ′ )+ cosh ρ (cos θ + cos θ ′ )(cos θ − cos θ ′ )(cosh ρ − cos θ )(cosh ρ ′ − cos θ ′ ) (cid:21) + ( x − y ) cosh ρ ′ cosh ρ ′ − cos θ ′ ! . Now note that cosh ρ ′ − cosh ρ = ∂ ω φ α and we already have expressions for cos θ − cos θ ′ and x − y involving combinations of ∂ ω φ and ∂ s φ .Hence we can write ( x − y )( ξ + η ) in the form( x − y )( ξ + η ) = f ( x, y, s ) ∂ s φ + ωf ( x, y, s ) ∂ ω φ. For future reference, note that our calculation in this section shows that(39) cosh ρ cosh ρ − cos θ − cosh ρ ′ cosh ρ ′ − cos θ ′ = (cosh ρ cosh ρ ′ + cos θ )(cosh ρ ′ − cosh ρ )(cosh ρ − cos θ )(cosh ρ ′ − cos θ ′ )+ cosh ρ (cos θ + cos θ ′ )(cos θ − cos θ ′ )(cosh ρ − cos θ )(cosh ρ ′ − cos θ ′ )A.5. Expression for ( x + y )( ξ − η ) . This is very similar to the derivation ofthe expression we obtained for ( x − y )( ξ + η ). ISTATIC SAR IMAGING 15
A.6.
Expression for ρ − η . Using (33) and (35), we have ξ − η = ω (cid:18) x | x − γ T | + x | x − γ R | (cid:19) − (cid:18) y | y − γ T | + y | y − γ R | (cid:19) ! = 4 ω (cid:18) x cosh ρ (cosh ρ − cos θ ) − y cosh ρ ′ (cosh ρ ′ − cos θ ′ ) (cid:19) = 4 ω ( x (cid:18) cosh ρ (cosh ρ − cos θ ) − cosh ρ ′ (cosh ρ ′ − cos θ ′ ) (cid:19) + ( x − y ) cosh ρ ′ (cosh ρ ′ − cos θ ′ ) ) . Now using the computations for x − y and ( x − y )( ρ + η ), in particular (39),we can write ξ − η in the form ξ − η = ωf ( x, y, s ) ∂ s φ + ω f ( x, y, s ) ∂ ω φ for smooth functions f , f . References [1] L-E. Andersson,
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