The Hyperbolic Geometry of SAR Imaging

Transcription

1 The Hperbolic Geometr of SAR Imaging Andrew S. Milman Northrop Grumman Airborne Ground Surveillance and Battle Management Sstems West NASA Blvd., Melbourne FL 39 April 1, Abstract This paper shows how we can use hperbolic functions to write an exact mathematical representation of SAR imaging. This problem is primaril a geometric one, that of accounting for curved wavefronts: the spirit of this paper is to emphasize these geometrical properties over electromagnetic ones. This gives us a new and fruitful wa to think about SAR imaging. Within this framework, I show how to correct for deviations of a radar from a straight flight path. This method will work even in situations where the curvature of the wavefronts is ver large, where traditional methods do not. The image-formation algorithm, called ω-k migration, that results from this analsis of SAR imaging is simpler and faster than polar formatting, especiall for radars with ver large beamwidths as the will at ver low frequencies. As an added benefit, ω-k migration is surprisingl simple to derive. 1. Introduction Historicall, processing snthetic aperture radar (SAR) data to produce images has used an approximation to express the instantaneous distance from the radar to a point on the ground. While this approximation is often adequate, it is still an approximation: we would prefer to have an exact expression both to aid our understanding of the SAR imaging problem and to improve the accurac of our computations. In this paper, I derive an exact mathematical representation of the SAR-imaging problem. We should be pleased to see that the result is ver simple. In particular, I shall show how, within this framework, we can make exact corrections for lateral motions of the radar. The usual SAR-imaging approximation ignores the curvature of the wavefronts: this leads directl to the now-familiar polar formatting (see Walker, 198; Jakowatz et al., 1996, provide a good discussion of SAR processing using polar formatting). When the SAR antenna has a narrow beamwidth, this approximation ma be adequate, but it breaks down when the beamwidth is large, as it will be for low-frequenc SAR s of the tpe that are being studied for foliage-penetrating applications. However, using hperbolic geometr, we can develop expressions that take the curvature into account exactl: no approximation is necessar. This leads to an alternative method of producing SAR images, which is called ω-k migration (see Milman, 1993, and references cited therein; Prati et al., 199; and Cafforio et al., 1989, 1991a and 1991b, are the original papers on the subject, as far as I know). Not onl is this method more accurate than polar formatting, it is also simpler and easier to implement on a computer. There are man advantages, and no known drawbacks, to using ω-k migration.

2 Hperbolic Geometr of SAR Imaging a) Man Subimages Signal Histor Signal Histor b) Single Image Figure 1. a) Forming several subimages; b) forming a single image. Polar formatting has the limitation that, because of the plane-wave approximation, the focus of an image deteriorates with increasing distance from the scene center. This requires that we divide the image into man sub-images; motion-compensate each pulse to the center of each sub-image; form these separate sub-images; and, finall, patch them together (Carrara et al., 1995b). I show this in Figure 1a. If there are N sub-images, the data need to be processed N times. Aside from the extra computation costs, this introduces discontinuities in the phase of the complex image; these discontinuities will affect algorithms that use this complex phase. We shall see that ω-k migration processing has none of these drawbacks. One wa to view migration processing is to consider, as I do here, the geometrical aspects of the problem, rather than concentrating on the electromagnetic formalism. Central to this issue is the question of how rotating the scene being imaged affects the signal histor since the scene appears to rotate as the SAR flies b it and how rotation affects its Fourier transform. Consider a function f(x,) and its Fourier transform F(r,s). A rotation of f in the x- plane produces the same rotation of F in the r-s plane (I discuss this further in section 5). If we can neglect the curvature of the wavefronts, we can use this propert of Fourier transforms to develop a polar-formatting algorithm (see Walker, 198). In polar formatting, the pulses are laid out in a radial manner, with range frequenc represented in the radial direction, going from f to f + B, the minimum and maximum frequencies in the chirp (i.e., B is the bandwidth of the radar; see Figure ). The direction changes to reflect the changing direction from the scene being imaged to the radar, so the angular coördinate φ in this diagram corresponds to the direction from the SAR to the scene center. The curvature of the lines of constant range frequenc, in this picture, depends on the ratio of the bandwidth B to the center frequenc f + ½B, and the total angle subtended b the SAR flight path. However, because the wavefronts are curved, the resulting image is onl properl focused at the center; the defocusing near the edges limits the size of the image that can be formed. In this discussion of migration processing, I shall start with the data as representing the reflected power as a function of distance from the SAR to the target (ρ) and of pulse

3 Hperbolic Geometr of SAR Imaging f +B f φ Figure. Polar formatting, showing the original data (solid lines), which form an angle φ corresponding to the angle subtended b the snthesized aperture, and f and B, which are the lowest frequenc in the chirped pulse and the bandwidth. Each pulse is plotted in a radial direction. The dotted lines are the data interpolated to the x coördinate sstem. number, or the position of the SAR along the flight track (t). In particular, I will neglect the fact that most radars are chirped i.e. the frequenc increases (or decreases) linearl during a pulse and assume that the radar in question transmits pulses with a single wavelength λ. In Milman (1993), I showed that the case of a chirped radar can be reduced to this simpler case. In practice, this is done b de-chirping the returned signal. (E.g., see Franceschetti and Lanari, 1999, Chapter 1.) Carrara et al. (1995a, pp. 41 ff) and Golden et al. (1994) discuss a similar method that the call range migration. Although it is based, in part, on Milman (1993), the do not make use of the propert that the migration formalism is an exact representation of the SAR imaging problem. This ma be, in part, because the have not adopted the geometrical outlook I espouse here. Franceschetti and Lanari (1999, pp and 74 ff) also emplo the migration method, in that the make use of the Stolt transform described below, but the do not make use of the formalism that I present here. In addition, Franceschetti and Schirinzi (199) and Franceschetti et al. (1996) both avoid the planewave approximation (which is equivalent to approximating the distance R to a target b a one-term Talor series), and the both use a stationar-phase approximation in creating SAR images. However, their work does not provide the geometrical insight that can be obtained from the method I describe here. Let me give an analog. Suppose that we were unaware of Newton s law of gravitation, but we knew that, given two bodies separated b the distance R + r, where r << R, the gravitational force between them is proportional to 1 r 3r +. R R R 3 4 3

4 Hperbolic Geometr of SAR Imaging It might be possible to calculate some approximate planetar positions from this, but without knowing the exact relationship, that the gravitational force is proportional to 1/(R + r), it would not have been possible to land a spaceship on the moon. For a similar reason, knowing this exact solution to the SAR-imaging problem is important for understanding it. There is alwas a need to correct for lateral motions of the radar, i.e., known deviations from a straight flight path. This problem is especiall severe in the case of lowfrequenc radars used for foliage-penetration applications, where the beamwidths can be ver large. In this paper, I show how to correct for these motions globall, in the sense that, assuming that we know the deviations from the nominal, straight flight path, we can perform a single correction to the data that compensates for that motion everwhere within the field-of-view of the radar. B contrast, the range migration algorithm discussed b Carrara et al. (1995a), Golden et al. (1994), and Carrara et al. (1995b) (and also personal communication) compensates for lateral motions b creating sub-images with the data compensated to the center of the scene (see Figure 1a). This onl accounts for the lateral motions of the radar approximatel and requires that the same raw data be processed man times, once for each sub-image that the signal histor contributes to. With ω-k migration, we can process the data once to produce the full scene, as shown in Figure 1b.. An overview of ω-k migration.1 Hankel Transforms. Before I start the derivation of ω-k migration, I should show ou where we re going. I want to show that there is a closed-form expression that relates the signal histor from the SAR to the distribution of targets on the ground, and that this expression has an analtical inverse. In so doing, I make use of Hankel transforms, which are similar in man was to the more familiar Fourier transform. I discuss Hankel transforms in more detail in Appendix A. We can write a Fourier transform as G ω = KF ωx g x d x, (1) where g(x) is an arbitrar function and the Fourier kernel K F (ωx) = exp{-πiωx}. This transform has an inverse g x = K * ωx G ω d ω, () where K F * is the complex conjugate of K F. Similarl, a Hankel transform is F 4

5 Hperbolic Geometr of SAR Imaging G ω = K ωx g x d x, (3) H where the Hankel kernel is K xh ω x, (4) H = and where H () (x) = J (x) iy (x) is a Hankel function of zero order of the second kind; J and Y are Bessel functions of the first and second kind. This transform also has an inverse: g x = K * ωx G ω d ω. (5) H Hankel transforms are less familiar than Fourier transforms but have similar properties. Now suppose I have a function A(x) whose Fourier transform is a(ω). If I measure a(ω), it is trivial to calculate A(x): we just use an inverse Fourier transform. Similarl, if B(x) is the Hankel transform of b(ω), we can calculate B from b with an inverse Hankel transform. We shall see later that we can use the usual Fast Fourier transform (FFT) software to calculate a Hankel transform, so it can be calculated as efficientl as a Fourier transform. Suppose that s(t,ρ) is the measured signal histor from a SAR, where ρ is distance from the radar to a reflector and t, the (slow) time that measures the position of the SAR along the flight track (here, I shall define the units so that the speed of the SAR, v = 1). Let S(ω,k) be the D Fourier transform of s(t,ρ); I shall show that S(ω,k ) is a D transform of the scene reflectivit σ (x,), where k = k - ω. The change of the independent variable from k to k is an important part of this theor. This transform is a Fourier transform in x, the along-track direction, and a Hankel transform in, the cross-track direction. Let me write the combined Fourier-Hankel kernel as Then I shall show that πiωx K = e H πk k F H. (6) S ω, k = K ωx, k σ x, d x d. (7) F H But if this is so, we can form a SAR image easil: the image is just * I x, = K ωx, k S ω, k d k d ω. (8) F H 5

6 Hperbolic Geometr of SAR Imaging SAR flight path (x - t) h ρ R x target Figure 3. SAR imaging geometr. This is an exact relationship: I have not had to make an plane-wave or similar approximation to arrive at equation (8). Furthermore, this relationship is valid regardless of the width of the size of the region illuminated b the antenna; there is no loss of focus at the edges. I am alwas surprised at how simple this solution is. It is customar to analze man kinds of SAR imaging problems b writing the signal histor as a Talor series and keep the first two or three terms. This is often an analtical nightmare, and it requires us to justif ignoring all terms higher than a certain order. But because ω-k migration provides an exact analtical relationship between signal histor S(ω,k ) and the scene σ(x,), we can analze SAR imaging problems quite easil within this framework. I provide one such analsis in this paper, pertaining to the question of how to correct for lateral motions of the SAR, but first, I shall provide a rigorous derivation of the results that I have claimed in this section.. Geometr and signal histor for a straight flight path Consider a radar mounted on an airplane that is used to make snthetic aperture radar (SAR) images of the earth s surface. Something that is well known, but apparentl often overlooked, is that the shape of the curve that describes the instantaneous distance from the SAR to a fixed point on the ground is an hperbola. We can make use of this propert to simplif the analsis of the SAR imaging problem; Figure 3 shows the relevant geometr. Here, h is the height of the radar above the (plane) surface of the earth. The radar moves with velocit v along the x-axis. For simplicit, I shall use units where v = 1; therefore, t is a distance (as well as slow time). Assume that the time t = corresponds to x=, so t measures the position of the SAR along the flight track. Furthermore, the flight path is assumed to extend from t to +t. I note, in passing, that I am making no assumption as to where the scene being imaged is in relation to the SAR: there is no restriction that it be at broadside. In the present discussion I treat the so-called spotlight mode, where the radar beam moves continuousl, relative to the radar, during the data collection so that the illumination remains fixed on the scene being imaged. Although it is customar to include an antenna weighting function w(x,) explicitl, I shall include it in the definition of the complex reflectivit σ to keep the notation simple. The extension to the stripmap mode, where the direction of the beam is fixed relative to the radar, is straightforward. 6

7 Hperbolic Geometr of SAR Imaging The distance from the SAR, located at the position (t,, h), to a point at (x,, ) on the surface is R = h + + x t. (9) We can reduce this to an equivalent -dimensional problem with the distance in the ground plane given b ρ = R h = + x t. (1) Then the signal histor is 4πiρ λ s t, ρ = e σ x, d s, (11) C where σ (x,) is the complex radar backscattering coefficient at location (x,); λ, the electromagnetic wavelength; and the curve C is the path defined b ρ = + (x - t). The complex exponential in this equation represents the relative phase of the reflected signal. The path of a point on the ground, in the frame of reference fixed to the airplane, is = ρ x t, (1) which, of course, is the equation for an hperbola. This suggests that we use hperbolic functions to describe the geometr of SAR imaging. I shall show how this allows us to use an exact analtic description of the SAR geometr. This will lead to an algorithm for producing SAR images that is mathematicall exact; this is, in part, a simplified derivation of the so-called ω-k migration algorithm described b Milman (1993) and others. The hperbolic functions sinhψ and coshψ are related b If we let cosh ψ sinh ψ = 1. (13) sinh ψ = x t, (14) then ρ = + x t = 1 + sinh ψ = cosh ψ. (15) Figure 4 shows the geometric meaning of the angle ψ. It shows the path of an object on the ground located at a distance from the SAR track, as seen from the aircraft. Now, since =ρsechψ, we can write x and in the form 7

8 Hperbolic Geometr of SAR Imaging ψ x-t x Figure 4. Hperbolic coördinates. x = ρ tanh ψ + t, (16) = ρsech ψ. (17) In order to develop an exact representation the phase histor using hperbolic functions, we will need to transform the variables of integration from (x,) to (ρ,ψ). We will need to use the relations tanh ψ + sech ψ = 1, d d tanh ψ = sech ψ and sech ψ = sech ψ tanh ψ. (18) dψ dψ The Jacobian of this transformation is ρsechψ. The complex signal histor is πiρ 4 1 λ s t, ρ = s t, ρ = e σ ρ tanh ψ + t, ρsechψ sechψ d ψ. (19) ρ The factor 1/ρ was introduced to cancel the factor ρ in the Jacobian. Now take the -dimensional Fourier transform of s(t,ρ): ( ) πik ( ρ+ωt) ( ) S ω, k = e s t, ρ dρdt 4πρ i ( ) λ πik ( ρ+ωt) e e tanh t, sech sech d dd t. () = σ ρ ψ+ ρ ψ ψ ρ ψ Transform this b letting = ρsechψ, so 8

9 Hperbolic Geometr of SAR Imaging ( ) ( ) π i ( k + / λ) coshψ+ωt. (1) S ω, k = e σ sinh ψ+ t, ddd t ψ Now substitute x = t + sinhψ, so ( ) ( ) πω i x π i ( k + / λ) coshψ ωsinhψ. () S ω, k = e e σ x, ddxdψ.3 Stolt and Hankel transforms Now make the following substitutions: let k "!# $% & = k + λ ω, k = k cosh φ, and ω = k sinh φ. (3) Note that the radial wavenumber ranges from /λ - ½ k to /λ + ½ k, where k = 1/δr, and δr is the spacing in distance. This substitution transforms k, the wavenumber in the radial direction, to k, the wavenumber in the -direction: this effectivel removes the curvature of the wavefronts. Using the relationship coshψcoshφ + sinhψsinhφ = cosh(ψ +φ) in equation (), we get the simple result that ' ( +, ) * i x S ω, k = e e σ x, dx d d ψ. (4) - π ω πik cosh ψ φ The transformation of S(ω,k) to S(ω,k ) is a one-dimensional interpolation, called the Stolt transform. Compare equation (38) in Appendix A, which is the integral representation of a Hankel function, to the integral over ψ in equation (4); this shows that equation (4) is a Hankel transform of the complex reflectivit σ. I describe the relevant properties of Hankel functions and Hankel transforms in Appendix A. Knowing this, we can perform the integration over t analticall, using equation (38): ' ( +, ' ( ) * πiωx S ω, k = H πk e σ x, dx d = - The inverse of equation (5) is - +, ' ( πiω t, π H k e σ ) * x dx d. (5) 9

10 4 4 4 Hperbolic Geometr of SAR Imaging. / πiωx σ x, = H πk e S ω, k k dk d ω. (6) We should note the factor exp{-πik cosh(ψ -φ)} in equation (5): to the extent that we can take the infinite limits of integration seriousl, the value of φ is immaterial. In realit, there is onl a finite angle ψ that we integrate over, so the value of φ will make a small difference. However, this is a limitation on our abilit to form images from a finite amount of data, and will arise no matter how we process the data: it is not a shortcoming of the ω-k migration method. This shows that, without making an plane-wave approximation, it is possible to write the radar cross-section distribution in terms of the signal histor. We can invert it b using a Fourier transform in the along-track direction, and a Hankel transform in the cross-track direction. The Hankel transform is unfamiliar, and (to m knowledge) no one has Fast Hankel Transform (FHT) software the wa we have FFT s. Fortunatel, we can use the approximation given b equation (39) in equation (6), which is accurate as long as πk > 3 (see Appendix A). So the Hankel transform we need to perform is, aside from division b (πk ) ½, equivalent to an FFT, and we can use the usual FFT software. The form of the Stolt transform given in equation (3) is different from the usual definition e.g., in Milman (1993). Here, I have used k + /λ, instead of just k, to account formall for the factor exp{-4πiρ/λ} in equations (11) et seq. We should also note (1) that we can arrive at the approximations for H in equation (39) b emploing the method of stationar phase (see Milman, 1993), and we do not need to mention Hankel functions at all if we do not want to. Carrara et al. (1995a) and Franceschetti and Lanari (1999) do just this. However, it is useful to see how we can represent the SAR imaging problem exactl using a Hankel transform, even if it does not directl affect the computational scheme for creating SAR images. 3. Shift to Scene Center As I have formulated this problem, the origin of the coördinate sstem is the center of the flight track. However, if we do not move the origin to the center of the scene being imaged, we increase the bandwidth of the signal enormousl. To this end, we can shift the coördinate sstem to the point (x, ) as follows. In equation (4), replace x and with x - x and - : ω, = σ ( +, + ) d d dψ πω i ( x x ) ( ) ( ) ( ) πi k cosh ψ φ S k e e x x x 4 i x i k i x ik cosh = e π ω e π e π ω e π ψ φ σ x, dx d d ψ. (7) 3. / 1

11 Hperbolic Geometr of SAR Imaging Signal Histor s(t,ρ) -D FFT S(ω,k) Shift the Origin πi ( ω x + k) ½ e k S ω, k ( ) Stolt Transform S(ω,k ) -D Inverse FFT Image I(x,) Figure 5. Steps in ω-k SAR processing. Here, I have used the approximation to the Hankel function given in equation (37), and used σ (x,) to denote reflectivit pattern shifted so that it is centered at (x, ). Carrara et al. (1995a) call this a matched filter, but I think that it is something of a misnomer. It is not a matched filter in the Wiener sense, but onl a shift of the coördinate sstem. This is a necessar step in digital data processing because it reduces the bandwidth needed in the along-track direction (efforts like those of Golden et al. to reduce the bandwidth are unnecessar). Finall, the image is found from the two-dimensional Fourier transform of S: πi( ω x + k) πi( ω x+ k) (, ) = 5 ( ω, ) dωd. (8) I x k S k e e k This comes about because we use equation (39) to approximate H () (πk ) in equation (6). The steps forming an image using ω-k migration are shown in Figure Limits of Integration Over ψ In the integral in equation (4), the limits of the integration over ψ are formall ±. In realit, the range of ψ is limited b the geometr of the data collection. However, these limits are obviousl finite. Suppose we are imaging a scene centered at location (x, ), with the distance to the center being ρ = (x + ) ½. If we use a stationar-phase analsis, it is clear that the integral ψ6 1 πik ψ φ ψ e cosh7 8 d ψ (9) is zero unless the stationar point of the integrand where ψ = φ is included in the interval (ψ 1,ψ ). This corresponds to the geometric requirement that 11

12 Hperbolic Geometr of SAR Imaging Range Wavenumber Range Wavenumber Azimuth Wavenumber (m) Figure 6. Extent of a point target in ω-k space. In this example, the target is at a boresight angle of (broadside). Azimuth Wavenumber (m) Figure 7. Same as Figure 6, except the boresight angle is 5. x t ω x + t < < ρ k + / λ ρ, (3) where ±t are the limits of the flight track. This follows directl from the definitions in equation (3). Therefore, the signal histor corresponding to a single point target onl occupies a restricted region of (ω,k) space. It is the extent of ω in this region that determines the sampling requirement for the SAR data collection. This is illustrated in Figure 6; note that the azimuth wavenumber ω is centered at, but the range wavenumber k is centered at /λ. For a target at, this region is centered at ω =. Figure 7 shows a target at a squint angle of 5. Suppose that the length of the SAR track extends from t to +t, and there are N pulses during the collection. Then the pulse spacing is δx = t /N, and the wavenumber ω is in the interval 1 1 ω < δx δx. (31) Similarl, if the range resolution is δ, the range wavenumber k is in the interval 1 1 k λ δr < δr + λ. (3) Equation () shows us that the radial wavenumber is centered at /λ. In general, ω is centered at x ω =, (33) λρ 1

13 > > Hperbolic Geometr of SAR Imaging k k ω λ/ ω c c r r η η d d x x Figure 8. The part of the ω-k plane observed with a scene at broadside (left) or squinted (right). Figure 9. If the SAR is displaced b an amount η from the x-axis to the x -axis, we should correct the measurements from the line c -d to c-d. Changing the phase of each pulse shifts the data radiall from r to r. where x is the azimuth of the scene center, and ρ the distance to the center of the flight track. The range of ω will be shifted b an amount that depends on the squint angle, as shown in Figure 7. In order to process the data efficientl, it is necessar to limit the processing to the part of the ω-k plane where S(ω,k) can be different from zero. So we will want to limit the processing to the region where /λ - ½ k < k < /λ + ½ k, and x /λρ - ½ ω < ω < x /λρ + ½ ω. One wa to do this is to shift the data in ω-k space along each line of constant k b an amount that depends on k. To this end, let β ω / (k + /λ), and shift S(ω,k) b the amount β(k + /λ). Then, instead of equation (), we would have 9 : < = ; : > πi k ρ + ω βk t S ω, k = e σ ρtanh ψ + t, ρsechψ sechψ dρdt d ψ, (34) where k = k + /λ. It is straightforward to derive the appropriate formula that corresponds to equation (6). Note that this is different from rotating the signal histor in the t-ρ plane. If we did that, we would not be able to solve equation (5) for σ. We can solve it, however, if we shift the data in the ω direction, as I showed here. 5. Correction for lateral motions We have alread seen how to correct for deviations in the vertical direction from a straight flight path. But what happens if the SAR drifts laterall from this straight path? Suppose that, at time t, the SAR is displaced laterall b an amount -η(t). This is equivalent of having the scene move in the -direction b η(t). If we simpl multipl each pulse b a complex phase to correct for these motions b shifting it toward or awa the SAR b 13

14 Hperbolic Geometr of SAR Imaging an amount -η, we are making an error because, while the SAR was displaced in the - direction, the correction is being applied in the radial direction see Figure 9. For widebeamwidth SAR s, this is a serious problem. We can perform the correction properl as follows. The central notion of this theor of SAR image formation is that equation (7) is exact, and that it has an exact mathematical inverse. As far as I can tell, onl this problem can be solved exactl. If the radar moves in an wa other than a straight line, no exact solution is possible. Therefore, to account for lateral motions of the radar, we need to break the flight path into straight segments. In general, each one will be at some angle α to the nominal flight path and we need to correct each segment of the signal histor for this tilt. To do this, we need to know how to treat a rotation and translation of the scene denoted b σ (x,). In general, if F(ω,k) is the -dimensional Fourier transform of some function f(x,), and we rotate f in the (x,) plane according to then the Fourier transform is rotated similarl: x cosθ sinθ x = sinθ cosθ, (35) ω cosθ sinθ ω = k sinθ cosθ k. (36) It is not possible to rotate the signal histor s(t,ρ) directl in the (t,ρ) plane. Instead, we can account for the rotation σ (x,), in the (x,) plane b calculating S(ω,k ) and then rotating it in the (ω,k ) plane. We have alread seen in equation (7) how a translation of the scene σ changes S(ω,k ). [The reader can verif that, if we rotate the signal histor in the (t,ρ) according to equation (35), it is no longer possible to solve equation ()]. We can follow these steps to correct for these lateral motions: 1. Divide the true SAR flight track into linear segments. Suppose the nominal flight line is the segment AZ, as shown in Figure 1. In each segment (AB, BC, and so forth) the scene being imaged has a different direction and distance, as compared with the corresponding section of the path AZ. The longer the segments are, the better the corrections will be.. For each segment determine the transformation that corresponds to moving that segment to the nominal flight path. 3. Use an FFT and the Stolt transform to calculate S(ω,k ). 4. Rotate and translate S(ω,k ). 5. Reverse the processes in step 4. This properl corrects each piece of signal histor for the deviation from the nominal path. 6. Combine these pieces to produce a single signal histor and process it to produce an image, as outlined in the last section. 14

15 Hperbolic Geometr of SAR Imaging Scene A B C D E Z x Figure 1. Divide the flight path into straight segments. Each segment is characterized b a different relative position of the scene being imaged. It should be clear that, forming S(ω,k ) from one segment of the signal histor, we will lose some of the information that we could have had if we were able to process the entire signal histor at once. Our abilit to account for the rotation of each segment of the flight path and to remove its effect is limited b the length of each segment, just as the azimuth resolution would be if we made an image from that segment alone. In m opinion, this loss of information is unavoidable: the image will be degraded to some extent if the radar does not fl in a straight line, no matter what we do. This degradation is inevitable; it is not due the ω-k migration algorithm and cannot be avoided b using some other algorithm. This is not to sa that the method outlined above is the onl wa, or even necessaril the best wa, to correct for lateral motions. However, given a particular set of circumstances, we can use the insight we gained from the ω-k migration paradigm to devise the most practical method for making these corrections. 6. Conclusion I have demonstrated here that there is an exact mathematical solution to the problem of forming a SAR image from the signal histor, at least in the case where the radar flies along a straight path. While this is useful in its own right for deriving a simple imaging algorithm, it is also important for understanding the SAR-imaging problem in general, as it gives us a framework for correcting for the problems that arise in real life. For example, there is a straightforward method for compensating for lateral motions of the radar. Like other aspects of ω-k migration processing, it is based on using the known, exact mathematical relationship between the scene being imaged and the signal histor. Although the hperbolic functions involved ma seem strange at first, viewing the SAR imaging process in terms of these functions simplifies both the basic imageformation algorithm and making corrections for lateral motions. In particular, the ω-k migration algorithm is simpler than polar formatting and does not have the limitations on image size that are imposed on polar formatting b the plane-wave approximation. 15

16 A A A A Hperbolic Geometr of SAR Imaging Appendix A. Hankel Functions We shall use the zero-order Hankel functions, which are defined as H (1) (z) = J (z) + iy (z) and H () (z) = J (z) - iy (z), where J and Y are the zero-order Bessel functions of the first and second kind. We shall use the integral representations H ( 1) iz cosht z = e ( ) iz cosht z = e πi d t, (37) d t. (38) Later, we will use the following asmptotic approximations to H (1) and H () : H H B B C 1 x x ei x ¼ π, π B B C x x e i x π π ¼. (39) These approximations for H (1) (x) and H () (x) are ver accurate for x >3. We shall see that this will alwas obtain for an plausible SAR. In the event that we need a more accurate asmptotic expansion, a more elaborate one is given below. We can write the inverse Hankel transform as follows. If F(r) is a function such r F r d r <, (4) then B GH I 1 F r H ur rdr H us udu = F s. (41) (This relationship is also true when H (1) (ur)h () (us) is replaced with C (ur)c (us) and C is either of the Bessel functions J or Y.). This is just a restatement of equations (3) and (5). A more accurate asmptotic expansion is the following: H J K L M J K QPR ST U L M L M N O n z e i z ½n ¼ π k c iz k = 1 O z n, k + π 1 + n 1 n z k =, (4) 16

17 ] Hperbolic Geometr of SAR Imaging where n is the order of the Hankel function and and c n, k = V W X W [ Y Z \ 4n 1 4n 3 4n k 1 k k! c n, = 1 (43) (Lebedev, 1965, p 1). We can calculate H () from this b noting that it is the complex conjugate of H (1). References Cafforio, C., C. Prati, and F. Rocca (1989): Snthetic aperture radar: a new application for wave equation techniques, Geophsical Prospecting, 7, Cafforio, C., C. Prati, and F. Rocca (1991a): Full resolution focusing of Seasat SAR images in the frequencwavenumber domain, Int. J. Remote Sensing, 1, Cafforio, C., C. Prati, and F. Rocca (1991b): SAR data focusing using seismic migration techniques, IEEE Tr. Aerospace & Electronic Sstems, 7, Carrara, W. G., R. S. Goodman, and R. M. Majewski (1995a): Spotlight Snthetic Aperture Radar, Artech House, Boston. Carrara, W., Tummala, S., and Goodman, R. (1995b): Motion compensation algorithm for widebeam stripmap SAR, Proceedings of the SPIE meeting Orlando, FL, Volume 487. Franceschetti, G., and G. Schirinzi (199): A SAR processor based on two-dimensional FFT codes, Trans. Aerospace & Electronic Sstems, 6, Franceschetti, G., and R. Lanari (1999): Snthetic Aperture Radar Processing, CRC Press, New York. Franceschetti, G., R. Lanari, and E.-S. Marzouk (1996): A new two-dimensional squint mode SAR processor, IEEE Trans. Aerospace & Electronic Sstems, 3, Golden, A., Jr., S. C. Wei, K. K. Ellis, and S. Tummala (1994): Migration processing of spotlight SAR data, in Algorithms for Snthetic Aperture Radar Imager, D. A. Giglio, ed, SPIE Volume 3, Jakowatz, C. V., Jr., D. E. Wahl, P. H. Eichel, D. C. Ghiglia, and P. A. Thompson (1996): Spotlight-Mode Snthetic Aperture Radar: a Signal Processing Approach, Kluwer Academic Publishers, Boston. Lebedev, N. N. (1965): Special Functions and Their Applications, tr. R. A. Silverman, Prentice-Hall, Inc., Englewood Cliffs, NJ. Milman, A. S. (1993): SAR Imaging b ω-k Migration, International J. of Remote Sensing 14, Prati, C., F. Rocca, A. M. Guarnieri, and E. Damonti (199): Seismic migration for SAR focusing: interferometrical applications, IEEE Trans. Geoscience & Remote Sensing, 8, Walker, J. L. (198): Range Doppler imaging of rotating targets, IEEE Trans. Aerospace and Electronic Sstems, AES-11,