Synthetic-aperture imaging through a dispersive layer

Transcription

1 INSTITUTE OF PHYSICS PUBLISHING Inverse Problems 4) 57 5 INVERSE PROBLEMS PII: S ) Synthetic-aperture imaging through a dispersive layer Margaret Cheney 1 and Clifford J Nolan 1 Department of Mathematical Sciences, Rensselaer Polytechnic Institute, Troy, NY 118, USA Department of Mathematics and Statistics, University of Limerick, Republic of Ireland Received 4 July, in final form January 4 Published 7 February 4 Online at stacks.iop.org/ip//57 DOI: 1.188/ ///1) Abstract This paper develops a method for forming a synthetic-aperture image of a flat surface seen through a homogeneous layer of a material that is dispersive, i.e., its wave speed varies with frequency. We outline first a simplified scalar model for electromagnetic wave propagation in a dispersive medium; the resulting equation could also be used for acoustics. We show that the backscattered signal can be viewed as a Fourier integral operator applied to the ground reflectivity function. The reconstruction method, which is based on backprojection, can be used for arbitrary sensor paths and corrects for the radiated beam pattern, the source waveform and geometrical spreading factors. The method correctly reconstructs the singularities such as edges) that are visible from the sensor. 1. Introduction This paper is motivated by the problem of synthetic-aperture radar SAR) imaging of forested areas. Most of the foliage-penetrating radar systems [9, 49, 5] are airborne and use frequencies in the range 1 MHz 1 GHz, which correspond to wavelengths between m and cm. At these relatively long wavelengths, models for wave propagation through foliage generally take the form of homogenized effective medium models [6, 15, 1, 48]. Some of the foliage-penetrating radar systems, moreover, are ultrawideband: the frequency band for the Swedish CARABAS system [49], for example, is 9 MHz, and for the GeoSAR system [5] is 7 4 MHz. Over such a wide relative bandwidth, some of the effective medium models predict a variation in the complex permittivity by as much as 5%. This raises the issue of how to account for the dispersion i.e., variation of propagation speed with frequency) in the image formation process. The goal of the paper is to address this issue. In this paper, we consider the effect of dispersion on the formation of syntheticaperture images. We use the simplest possible example of a dispersive layer, namely a flat, homogeneous, dispersive layer over a flat surface. Ultimately, of course, this is much too simple to model real foliage, but it provides a beginning for the study of imaging through a dispersive medium. In particular, the theory developed in this paper could also be applied to imaging through materials such as soil [1, 51] and human tissue [1] /4/576$. 4 IOP Publishing Ltd Printed in the UK 57

2 58 M Cheney and C J Nolan Figure 1. Data collection geometry. The star represents a feature in the scene to be imaged, and the region between the planes is filled by a homogeneous dispersive material. For this simple model, we are able to obtain an explicit imaging formula and a corresponding point spread function that can be used to evaluate the image. Extensions to more complicated known) geometry can probably be made by combining the approach of [5] with geometrical optics for dispersive media [7]. The paper begins in section with a detailed discussion of a simplified scalar mathematical model for the received signal. This is followed, in section, by analysis of the received signal under certain general conditions on the nature of the dispersion. In particular, we show that the received signal can be expressed as a Fourier integral operator FIO) applied to the scene. For this FIO, we find, in section 4, an approximate inverse; this inverse provides the imaging formula. The paper concludes with some appendices containing technical details.. The mathematical model Our goal in this section is to derive a mathematical expression for the received radar signal. We model the scatterers of interest as variations in the ground reflectivity function supported on a plane. For scattering from this plane we use a single-scattering Born) approximation. The dispersive layer is modelled as a known) homogeneous dispersive half-space in which the plane of unknown reflectivity is embedded. This model ignores multiple scattering between the plane ground) and the top of the dispersive layer the foliage crown); we assume such effects have been subtracted out of the data. We assume that the sensor antenna) moves along a path γ that is well separated from the top at x = H ) of the dispersive half-space. We assume that the path is smooth; it is otherwise arbitrary, and for example would be allowed to perform loops and self-intersections see figure 1). Waves emanate from the sensor and are received back at the same sensor. We make the start stop approximation, i.e., we neglect motion of the sensor during the measurement, and we neglect issues related to the discreteness of the pulses. Derivation of the mathematical model involves 1) a scalar model for wave propagation, ) a model for the field emanating from an antenna, ) a model for the wave after it propagates through a homogeneous dispersive layer and 4) a model for scattering from the earth..1. The equations For SAR, the correct model is Maxwell s equations, which we write as E = t B, 1)

3 Synthetic-aperture imaging through a dispersive layer 59 H = J t D, ) D = ρ, ) B =. 4) Here E is the electric field, H the magnetic field, D the electric displacement, B the magnetic induction, J the current density and ρ the charge density. The four fields E, D, H, B are related by constitutive relations, which in this paper we assume to be of the form [5] Bt, x) = µ Ht, x), 5) Dt, x) = εs, x)et s, x) ds ε t E)t, x). 6) Here µ is the magnetic permeability of free space, which means that we are considering non-magnetic materials. The relation between D and E is a causal convolution in time with the electric permittivity ɛ. In general, the permittivity kernel is of the form εs,x) = ɛ x)δs) χs,x) with χ being smooth; ɛ is called the instantaneous response or optical response and χ is called the susceptibility kernel. Discontinuities in the electric and magnetic fields propagate with speed cx) = 1/µ ɛ x)) 1/. If we use 5) and 6) to eliminate B and D in 1) and ), and then substitute the curl of 1) into), we obtain E = t µ J tt µ ε t E). 7) Finally, we use the identity E = E) E to write 7)as E tt µ ε t E) E) = µ t J. 8) Up to this point, our only assumptions have been 5) and 6); we now make a simplifying assumption to reduce 8) to three uncoupled scalar equations. Assumption 1. We assume that E) = ; this reduces 8)to E tt µ ε t E) = µ t J. 9) Assumption 1 does not hold in general; instead, from ) and 6) we should have ε t E) = ρ. However, 9) holds within a homogeneous medium in which there are no free charges, and in particular it holds for wave propagation in air and within a homogeneous dispersive layer. In general 9) does not hold at interfaces between materials; at such interfaces there is coupling between the different components of E []. By using 9), we are ignoring such coupling, and we are thus ignoring polarization effects. Consequently we consider only one component of 9), thus reducing the problem to a scalar one. Because electromagnetic waves are rapidly attenuated in the earth, we assume that the non-dispersive) scattering takes place in a thin region near the surface. We use the superscript to designate points on the earth s surface, e.g. x T,x ) and zt,z ). Assumption. The permittivity kernel is of the form εs, x) = Vx T )δ x x) δs) ɛ ε r s, x), where x T = x 1,x ), where ε r and x are known, ɛ denotes the permittivity of free space and where V is unknown. The dispersive layer extends to x = H > x. We assume that the sensor path γ is well above x = H. Here V,theground reflectivity function, is the quantity we wish to reconstruct. The case in which the earth s surface is not flat was considered, for the non-dispersive case, in [7], which used techniques similar to those used in this paper.

4 51 M Cheney and C J Nolan With the notation of assumption, we write our wave equation as the single scalar equation E tt c ε r t E ) V δ) tt E = j s, 1) where c = 1/µ ɛ ) 1/ and where j s denotes one component of µ t J in 9). In 1), ε r is assumed known but V is unknown... Sources in and above a dispersive half-space In this subsection, we obtain an explicit representation for the half-space Green s function g, which satisfies gt, x, y) tt c ε r t g ) t, x, y) = δt)δx y). 11) Our derivation follows the appendix of [8]. We take the Fourier transform 11)int, which results in k ɛ r ω, x ))Gω, x, y) = δx y), 1) where we have written k = ω/c and ɛ r = ɛ/ɛ with ɛ being the temporal Fourier transform of ε. We consider the case in which { 1, for x >H ɛ r ω, x ) = ɛr ω), for x 1) <H. Because the medium does not vary in the x 1 and x directions, G is of the form Gω, x, y) = Gω, x 1 y 1,x y,x,y ). Finally we take the Fourier transform 1) also in the horizontal variables x 1 and x, which results in x k η ω, η T,x ) ) Ĝω, η T,x,y ) = δx y ), 14) where we have written the variables dual to x 1 and x in scaled form as kη T = kη 1,kη ), and where η ω, η T,x ) = ɛ r ω, x ) η T. Equation 14) can be solved explicitly [8] in our half-space case 1). In particular, away from x = y, the general solution of 14) is,forx >H, Ĝ ω, η T,x,y ) = A ω, η T,y ) e ikη x H) B ω, η T,y ) e ikη x H), 15) and, for x <H, Ĝ ω, η T,x,y ) = A ω, η T,y ) e ikη x H) B ω, η T,y ) e ikη x H), 16) where { 1 ηt η ω, η, for 1 > η T T ) = isgn k) 17) η T 1, for 1 < η T, and η ω, η T ) = ɛr ω) η T, 18) the branch of the square root being chosen so that the imaginary part of η is positive when ω and negative when ω<. For η T < 1,η is real; if η T > 1, η is purely imaginary. This latter case gives rise to evanescent waves, which decay exponentially in the direction of propagation. We note that η can be complex even for small η T. We define the vectors η ± = η T,η ± ), which satisfy η± η ± = ɛ r. The coefficients A ± and B ± of 15) and 16) depend on whether the source height y is greater than or less than H and on whether x is greater than or less than H and y. When x is

5 Synthetic-aperture imaging through a dispersive layer 511 greater than both H and y, the condition that ĝ be upgoing implies that B is zero; when x is less than both H and y, the condition that ĝ be downgoing implies that A is zero. Ĝ and its x derivatives are continuous except at x = y, where Ĝ is continuous but its x derivative jumps by minus one. Solving for the A and B in both cases results in the following. For y >H,wehave[8] { Ĝω, η T,x,y ) = i R η,η ) e ikη x y H) e ikη x y, for x >H kη T ) 19) η,η e ikη y H) e ikη H x), for x <H, and for y <H,wehave Ĝω, η T,x,y ) = i kη where and { T η,η ) e ikη H y ) e ikη x H), for x >H e ikη x y R η ),η e ikη x y H), for x <H, ) Tλ 1,λ ) = λ λ 1 λ, 1) Rλ 1,λ ) = λ λ 1. ) λ 1 λ Note that since the imaginary parts of kη and kη are non-negative, the exponents in 19) and ) are decaying. In order to model the field received at the antenna, we need the field transmitted through the layer from a source the antenna) above the layer, and the field from a source due to scattering) on the ground that is transmitted through the layer and received at the antenna. We denote these components by Ĝ and Ĝ ; more precisely they are the second line of 19) and the first line of ), respectively. The resulting frequency-domain Green s functions are G ω, x, y) = 1 it η,η ) π) kη e ikη T x T y T ) e ikη y H) e ikη H x) k d η T, ) G ω, x, y) = 1 ) it η,η π) kη e ikη T x T y T ) e ikη H y) e ikη x H) k d η T. 4) The time-domain Green s function is recovered from its Fourier transform by gt, x, y) = 1 e ikη T x T y T ) iωt Ĝω, π) η T,x,y ) dωk d η T. 5) Thus, for example, the field at the point x on the earth s surface due to a point source at y above the dispersive layer is g t, x, y) = 1 it η,η ) π) kη e ikη T x T y T ) iωt e ikη y H) e ikη H x) dωk d η T, 6) wherewehaveused). If we add and subtract the quantity η T x T in the phase of 6), and use the notation η ± = η T, η ± ) and x H = x T,H), we see that 6) involves a product of the plane waves exp[i kη xh y)] and exp[i kη x xh )], which correspond to propagation in the upper medium from y to the intermediate point x H and propagation in the lower medium from x H to x, respectively. We note that models with more layers can be accommodated by using a multiple-layer Green s function [4].

6 51 M Cheney and C J Nolan.. The field emanating from an antenna In the preceding section, we have obtained an explicit expression for the Green s function for a dispersive half-space. This Green s function corresponds to a source of the form δt)δx y). The antenna, however, is not a point source [5] δx), and the signal sent to the antenna is not a delta function δt). Therefore to obtain a simple antenna model, we replace δt)δx) of 11)byj s t, x) of 1). Typical antennas used for foliage-penetrating radar are horns, microstrip arrays [45], broadband dipoles [] and log-periodic arrays of dipoles [17]. For the case of dipoles, j s can be taken proportional to the time derivative of ) the current density on the dipole; for a horn or slotted waveguide, j s is often taken to be an effective current density over the aperture. We write j s in terms of its Fourier transform J s : j s t, x) = 1 e iωt J s ω, x) dω, 7) π where ω denotes the angular frequency. In practice, the waveform j s is such that only a certain interval [ω min,ω max ] contributes significantly to 7); we call this set the effective support of J s. The difference ω max ω min ) is the angular-frequency) bandwidth. The fact that j s is bandlimited means that ultimately we reconstruct bandlimited approximations to singular components of the coefficient V. The field E in emanating from the antenna then satisfies E in t, x) tt c ε r t E in) t, x) = j s t, x), 8) so that E in t, x) = g t t, x T y T,x,y )j s t, y) dt dy = e iωt G ω, x T y T,x,y )J s ω, y) dω dy. 9) We write y = γs) q, where γs) denotes the centre of the antenna. Then, using ) in 9), we have E in t, s, x,s)= 1 it η,η ) π) kη e ikη T x T γ T s)) iωt e ikη γs) H) e ikη H x ) J s ω, s, η T ) dωk d η T, ) where J s ω, s, η T ) = e ikη T q T η q ) J s ω, γs) q) dq, 1) is an approximation to the antenna beam pattern. We note that in reality, the antenna beam pattern is strongly affected by the vector nature of the current and the vector structure of Maxwell s equations. Approximation 1) can be used to give a good approximation to the size of the main lobe of the beam pattern, but does not necessarily provide an accurate model for the overall beam pattern..4. A linearized scattering model We write E = E in E sc in 1) and use 8) to obtain E sc tt c ε r t E sc) = V δ) t E. )

7 Synthetic-aperture imaging through a dispersive layer 51 We recall that the reflectivity function V contains all the information about how the medium differs from the background. It is V, or at least its discontinuities and other singularities, that we want to recover. We can write ) as an integral equation [] E sc t, x) = g t τ,x T z T,x,z )V z T )δ z z) τ Eτ,z) dτ dz. ) A commonly used approximation [1, 8], often called the Born approximation or the single-scattering approximation, is to replace the full field E on the right-hand side of ) and ) by the incident field E in, which converts )to E sc t, x) EB sc t, x,s) := g t τ,x T z T,x,z )V z T )δ z z) τ E in τ, s, z,s)dτ dz, 4) where the subscript B reminds us that we are using the Born approximation. The value of this approximation is that it removes the nonlinearity in the inverse problem: it replaces the product of two unknowns V and E ) by a single unknown V ) multiplied by the known incident field. The Born approximation makes the problem simpler, but it is not necessarily a good approximation. Another linearizing approximation that can be used for reflection from smooth surfaces is the Kirchhoff approximation, in which the scattered field is replaced by its geometrical optics approximation at the surface of the scatterer [5, 8]. Here, however, we consider only the Born approximation. In summary, we assume Assumption. The data S are linearly related to V, i.e., we use a single-scattering Born) approximation. In 4) we substitute ) and 4) with η replaced by µ) and simplify the result by carrying out the τ integration and one of the ω integrations: ) EB sc 1 it µ t, x,s)=,µ π) 5 kµ e ikµ T x T z T ) iωt e ikµ H z) e ikµ x H) Vz T )δ z z ) ω it ) η,η kη e ikη T z T γ T s)) e ikη H z ) e ikη γ s) H) J s ω, s, η T ) dωk 4 d µ T d η T dz, 5) where µ ± are defined as in 17) and 18). We note that physically, the full scattered field involves a reflection from the top of the foliage as well as a reflection from the ground beneath the foliage. This foliage-crown bounce is not included in E sc because it is part of the assumed known) incident wave. In other words, we are assuming that the foliage-crown bounce has been subtracted out; E sc involves only scattering from the ground..5. The received signal We model reception of the field 5) by the antenna by writing x = γs) q in 5), and calculating St,s) = EB sc t t, γs) q,s)j r t, γs), q) dt dq, 6) where j r denotes the reception pattern for the antenna. For an array antenna, for example, j r would be the possibly complex) weighting with respect to which the signals from the different

8 514 M Cheney and C J Nolan elements are combined. Generally the reception pattern is the same as the transmission pattern j s. We obtain St,s) = 1 ) T µ,µ π) 5 kµ e ikµ T γ T s) z T ) iωt e ikµ H z ) e ikµ γ s) H) Vz T ) J r ω, s, µ )ω T ) η,η kη e ikη T z T γ T s)) e ikη H z ) e ikη γ s) H) J s ω, s, η T ) dωk 4 d µ T d η T dz T, 7) where J r is defined in the same way as J s. Technical difficulties arise from evanescent waves, for which η ± or µ± is purely imaginary, and from horizontally propagating waves, for which η ± = orµ± =. The contribution to the data from evanescent waves, however, is negligible, because assumption implies that terms such as exp i kη z H )) are exponentially small. The contributions from horizontally propagating waves are also negligible, because such waves cannot travel between the antenna and the scatterer. Assumption 4. We neglect contributions from evanescent and horizontally propagating waves. We therefore insert into 7) a smooth cutoff function ψω,η T, µ T ) that restricts the domain of integration to be strictly within the set { η T, µ T < min{1,ɛ r ω)}}. The cutoff ψ must be chosen so that all its derivatives decay faster than any polynomial. The idealized inverse problem is to determine V from knowledge of S for t T 1,T ) and for s on some interval s min,s max ). A number of technical difficulties arise if we attempt to image points directly underneath the antenna [7]. In particular, we will see that our imaging algorithm cannot be used for data coming from locations directly underneath the current location of the antenna. We therefore make the following assumption. Assumption 5. At every position γs) on the smooth) flight path, the height γ s) and the time T 1 s) at which data recording begins are related by T 1 s) > T > γ s) x / c for some T. The abrupt ends of the curve γ tend to cause artefacts in the image; consequently, it is useful to multiply the data by a smooth taper function ms, t) which is zero outside s min,s max ) T 1,T ). Consequently, we write the data as dt,s) = ms, t)st, γs)) = ms, t) π) 5 ) it µ,µ kµ e ikµ T γ T s) z T ) iωt e ikµ H z ) e ikµ γ s) H) Vz T ) J r ω, s, µ ) it ) η,η kη e ikη T z T γ T s)) e ikη H z ) e ikη γ s) H) J s ω, s, η T )ψω, η T, µ T )ω dωk 4 d µ T d η T d T z. 8). Analysis of the received signal Our goal in this section is to write the model for the data 8) in the form of an FIO [1, 19, 44] and then use FIO theory to analyse it. In order to do this, we need certain conditions on the

9 Synthetic-aperture imaging through a dispersive layer 515 dispersion properties of the intermediate layer. These conditions are satisfied by the Debye and Lorentz models, and also by some, but not all, dispersion models developed specifically to model foliage. Assumptions about the dispersion are necessary because the model for the data 8) is not presently written in the form of an FIO: the phase involves a complicated dependence on the internal variables, namely the temporal frequency ω and the scaled spatial frequencies µ T and η T. The phase of an FIO must be homogeneous of degree 1 in these variables. For the scaled spatial frequency variables µ T and η T, there is an easy remedy: we make the change of variables η T = η T /ω and µ T = µ T /ω. The dependence on ω, however is more complicated: ω occurs in the functions µ and η through their dependence on ɛ rω). The dependence of ɛ r on ω is not arbitrary: from 6) we know that the Fourier transform of ɛ r is zero on the negative real time) axis, and therefore by the Paley Wiener theorem [4], ɛ r must be analytic in the upper half-plane. Moreover, smoothness in the time domain implies large-ω decay of ɛ r ɛ of at least Oω 1 ) []. This latter fact enables us to simplify the phase of 8) by reassigning parts of µ and η to the amplitude. Assumption 6. We assume that ɛ r ω) is of the form ɛ r ω) = ɛ ɛ s ω), 9) where ɛ s ω) satisfies ω β ɛ sω) C β 1ω ) 1 β )/, 4) for every non-negative integer β. We give examples below of specific models, some of which satisfy this assumption and some of which do not. The large-ω decay of ɛ s implies that ɛ s appears only in the remainder terms of large-ω asymptotic calculations. In order to exploit this fact, we write η ω, η T ) = ɛr ω) η T as η ω, η T ) = η η T ) η a ω, η T ), 41) where η η T ) = ɛ η T and η a ω, η T ) = η ω, η T ) η η T ). 4) We define µ a similarly. We will include the remainder term exp i kη a ω, η T ) z H )) in the amplitude, thus removing it from the phase of 8). This enables us to replace η and µ, which depend on ω, byη and µ, which do not..1. Specific models for dispersion Examples of models that do and do not satisfy assumption 6 are given below. Highfrequency decay rates must be faster than ω 1 for time-domain continuity [], but some of the phenomenological models do not satisfy this condition Fung Ulaby model 8 17 GHz). The model of Fung and Ulaby [15] for the effective permittivity of leafy vegetation is based on a model for the permittivity of the leaf material together with a mixing model.

10 516 M Cheney and C J Nolan If ν is the fractional volume occupied by leaves, then the effective relative permittivity of the foliage is ɛ r ω) = ɛ a 1 ia 1b ω, 4) where ɛ = 5.5ν 1 ν), a 1 = 51.56ν w.5)ν with ν w being the water volume fraction within a typical leaf, a = a 1 b, and b is an empirical factor involving the temperaturedependent relaxation time for water. At C, we have roughly b = 185/πc ). The Fung Ulaby model satisfies assumption Brown Curry model 1 MHz 1 GHz). The Brown Curry model [6] is a model for a sparse random medium such as the branches and tree trunks of a forest. A modification due to Ding [11] of the Brown Curry model is ɛ r ω) = [ ɛ b 1ω/ω c ) ] [ ω/ωc )b i 1ω/ω c ) ] a. 44) πω).96 Here ɛ 4.5,bis a constant that depends on the density and fractional volume of wood, a = respectively ) when the electric field is parallel perpendicular) to the wood grain and πω c is a temperature-dependent frequency that is roughly GHz at 5 C. The Brown Curry model does not quite satisfy assumption 6, because the last term of 44) decays slightly more slowly than ω Debye model. The Debye model is generally good for polar molecules such as water in the microwave regime. The Debye model is ɛ r ω) = ɛ ɛ s ɛ 1 iωτ, 45) where for water, typical values are as follows. The zero-frequency relative permittivity is ɛ s = 8.5, the infinite-frequency relative permittivity is ɛ = 1. and the relaxation time is τ = 8.1 ps [41]. The Debye model satisfies assumption Lorentz model. The Lorentz model, which is a harmonic-oscillator model, is generally good for solid materials. The permittivity for a Lorentz medium is given by b ɛ r ω) = ɛ ω ω, 46) iωδ where ɛ is real and positive. The Lorentz model satisfies assumption 6... The forward operator is an FIO We write 8)asthemapF : V d. We write F in the form of an FIO by making the change of variables η T = η T /ω and µ T = µ T /ω and by moving ηa and µ a from the phase into the amplitude: FVs,t) = e iφt,s,x,ω, µ T, η T ) At, s, x,ω, µ T, η T )V x T ) d µ T d η T dω d x T, 47)

11 Synthetic-aperture imaging through a dispersive layer 517 where φt,s,x,ω, µ T, η T ) = ω t [ µt ω γ ) T s) x T ) µ H x µ γ s) H) ) ]/c, 48) η T ω x T γ T s)) η H x ) η γ s) H) where µ = µ ω, µ T /ω), and η, µ and η are defined similarly, and ms, t) T µ At, s, x,ω, µ T, η T ) =, ) µ π) 5 k µ e ik µ a H x ) J r ω, s, µ T /ω) T η, ) η k η e ik η a H x ) J s ω, s, η T /ω)ψω, η T /ω, µ T /ω). 49) The following proposition shows that moving η a from the phase to the amplitude is legitimate, because the resulting amplitude satisfies the conditions for being in the symbol class S of order and type 1, )). Proposition 1. For every triple of non-negative integers β = β 1,β,β ), ãω, η T ) = aω, η T /ω) = ψω, η T /ω) e ik η aω, η T /ω)h z ), 5) satisfies β 1 η 1 β η β ω ãω, η T ) C β 1 ω, η T ) ) β / = C β 1ω ) β / ; 51) i.e., a is in S. A similar estimate holds if η T is replaced by µ T. The proof of this proposition is given in appendix A. We now make an assumption about the antenna beam patterns. This assumption is satisfied, for example by a broadband antenna i.e., J r and J s are approximately independent of ω over the effective bandwidth). Mathematically, this assumption ensures that F is an FIO. Assumption 7. The antenna activation patterns J r and J s each satisfy, for some m, α s β ω β η Jω,s, η T ) C1ω η ) [m β β )]/, 5) sup s K where β η = β 1 η 1 β η, η = η 1 η, β =β 1 β, where K is any compact subset of R ω s min,s max ) R η, and the constant C depends on K,α,β,β 1,β and β. In other words, we assume that both the J s are in the same symbol class S m. assumption can be weakened to admit other symbol classes [1, 19]. With this assumption we have the following estimate. Proposition. Suppose assumptions 1 7 are satisfied. Then for some m, α s s sup s,t,x) K α t t α 1,α ) x 1,x ) β ω β 1,β ) η 1, η ) β,β 4 ) µ 1, µ ) At, s, x,ω, µ T, η T ) C1 ω η µ ) m β )/, 5) where β = 4 i= β i, α = α s α t α 1 α, where K is any compact subset of R t s min,s max ) R x T R ω R µ R η, and the constant C depends on K and on the α and β. Proof. A is a product of symbols and is thus again a symbol [19]. The only potential difficulty is zeros of µ ± and η±, but these have been excluded by assumption 4 via the smooth cutoff function ψ. This

12 518 M Cheney and C J Nolan Corollary 1. Under assumptions 1 7, F is an FIO. Proof. The phase function is homogeneous of degree 1 in the vector of internal frequency ) variables ω, µ T, η T ). The phase function is real valued, and on the effective support of the amplitude A, φ/ t = ω never vanishes... Analysis of the forward operator..1. The critical set. The main contributions to F come from the critical points of the phase, i.e., from the set of points s,t,x,ω, µ T, η T ) satisfying [ µt = ω φ = t ω γ ) T s) x T ) µ H x µ γ s) H) η ) T ω x ) T γ T s)) η H x η γ s) H) ]/c ω µi φ ηi φ ), 54) i=1 = µi φ = 1 c [ γ i s) x i ) µ i/ω = ηi φ = 1 c [ x i γ i s)) η i/ω µ H x ) µ i /ω µ η H x ) η i /ω η ] γ s) H), 55) ] γ s) H), 56) for i = 1,. We note that 55) and 56) imply that the term with the summation sign in 54) vanishes. Geometrical interpretation of the critical set. We will see below that equation 54) says that the travel time t is the sum of the travel times along the upgoing and downgoing paths from γs)to x. The equations for these paths are equations 55) and 56). Equations 55) and 56) can be written as γ 1 s) x 1 = γ s) x µ 1 µ = γ s) H µ γ 1 s) x 1 = γ s) x η 1 η = γ s) H η H x µ, H x η. We will see below that these equations express Snell s law for a ray refracting through the interface at x = H. A line with direction µ through the point γs) will intersect the plane x = H at a certain point see figure ), which we denote by x H = x H 1,xH,H). We write µ = µ 1,µ,µ ), µ = µ 1,µ, µ ) and similarly for η. Proposition Geometrical interpretation of 57)). On the critical set, µ = η = x H γs)) and µ = η = ɛ x x H ). The first and second lines of 57) therefore describe the same refracting ray path. The second equation of 57) therefore corresponds to a ray that starts at γs) with direction vector η = η 1, η, η ), travels in direction η until it hits the interface x = H at the point x H, and continues, now with direction vector η, to the point x on the surface x = x.the first equation of 57) corresponds to a ray that traces the same path in the opposite direction. 57)

13 Synthetic-aperture imaging through a dispersive layer 519 γ s) x = H x H µ x = x x µ Figure. The geometry of the critical set. On the critical set, µ = µ, η = µ,and η = η = µ. Proof of proposition. The first equation of 57) contains two pieces of information: first, the equation γ 1 s) y 1 µ 1 = γ s) y µ = γ s) H µ H y µ, 58) describes a geometrical set of points y = y 1,y,y ). Second, 57) says that the point x = x 1,x,x) lies in this geometrical set. First we consider the set of points y satisfying 58). If y = y H = y1 H,yH,H),wesee that 58) reduces to γ 1 s) y1 H = γ s) y H = γ s) H µ 1 µ µ, 59) which, if we denote the common ratio of 59) byρ H, can be written in vector form as y H = γs) ρ H µ. This says that a point y H = y1 H,yH,H) lies on the line with direction vector µ through γs). Since µ µ = 1, and, by assumption 4, µ is real, µ must be equal to γs) yh or its negative. In fact, since the third components of γs) yh and µ are both positive, we must have µ / = γs) y H /. If we now add and subtract y1 H µ1 and y H µ in the first and second members of 58), respectively, we obtain γ 1 s) y H 1 µ 1 yh 1 y 1 µ 1 = γ s) y H µ yh y µ = γ s) H µ H y µ ; 6) subtracting 59) from6) results in the equations for a line with direction vector η joining y H to a point y: y1 H y 1 = yh y = H y ; 61) µ 1 µ µ or, if we denote the common ratio of 6)byρ and carry out the subtraction of 59)from6), we see we can write 61) in vector form as y = y H ρ ρ H )µ. Taking y = y = ) y1,y,y in 61), where y is a point on the ground, shows that µ = ɛ yh y ). The same arguments apply to the second line of 57), except that now the third components of η and η are negative. Thus η = γs) y H ) and η = ɛ y H y ).

14 5 M Cheney and C J Nolan Proposition implies that we can write the phase as φt,s,x,ω, µ T, µ T ) = ω t [ µ γs) x H ) µ x H x) η xh γs)) η x xh ) ]/ ) c = ωt γs) x H ɛ x H x )/c ). 6) From 6), the interpretation of the phase in terms of travel time is clear. Proposition 4. Under assumption 4, equations 55) and 56)) can be solved for µ T and η T ) in terms of γs) and x. Proof. First we note that if we write µ T in polar coordinates, solving 55)forµ really requires only solving for the single variable r = 1/. µ 1 ) µ This we see from writing the first line of 57)intheform H ) x, 6) µ γ s) H γ j s) x j = µ j µ for j = 1, and dividing the j = equation by the j = 1 equation. We find that tan θs,x) := γ s) x 1 = µ. 64) γ 1 s) x µ 1 Thus knowledge of s and x determines θ. Formula64) determines θ up to an additive factor of π; knowledge of the positions of γs) and x determine the correct quadrant for θ. We write µ T = r cos θ,r sin θ), γ T s) x T = R cos θ,rsin θ). 65) With this notation, we have µ = 1 r ) 1/ and µ = ɛ r ) 1/, and each equation of 6) can be written as R r = γ s) H 1 r We rewrite this equation as = fs,x,r):= H x ɛ r. 66) r γ r ) s) H) H x 1 r ɛ r Rs, x), 67) where Rs, x) = γ T s) x T. The implicit function theorem guarantees that we can solve 67)forr in terms of s, x 1 and x whenever f/ r. We calculate f r = γ s) H [1 r ] ɛ H x / [ɛ r ] = γ s) H / ) µ H x ɛ µ. 68) The argument for 56) is similar. Proposition 4 says that the takeoff direction µ = η of the refracting ray is determined by its initial position γs) and its final position x. This is intuitively clear from Snell s law.) We denote by µ T s, x) and correspondingly rs,x) and θs,x)) the solution of 55) determined by proposition 4. We note that an explicit formula for µ T in terms of s and x could be obtained by converting Snell s law which is encoded in the definitions of µ and µ ) into a fourth-order polynomial for the coordinates of x H. Although an explicit formula for µ T in terms of s is complicated, a formula for µ T / s needed in the next section) can be obtained by implicit differentiation.

15 Synthetic-aperture imaging through a dispersive layer 51 Corollary. µ 1 s µ s = cos θ = sin θ r µ r µ γ s) γ 1 cos θ γ sin θ) γ s) H H x r R sin θ γ 1 sin θ γ cos θ), ) µ ɛ µ γ s) γ 1 cos θ γ sin θ) γ s) H H x r R cos θ γ 1 sin θ γ cos θ). ) µ ɛ µ 69) Here θ = θs,x) and r = rs,x). Proof. From the chain rule applied to 65), we have µ 1 s = µ 1 r r s µ 1 θ f/ s = cos θ r sin θ 1 θ s f/ r R γ 1 sin θ γ cos θ), µ s = µ r r s µ θ f/ s = sin θ r cos θ 1 θ s f/ r R γ 1 sin θ γ cos θ), where in the second equality we have used the implicit function theorem applied to 67) to obtain r/ s and applied to 65) as follows to obtain θ/ s. In particular, we consider the second equation of 65) to be two scalar equations that determine R and θ as functions of s with x fixed). Implicit differentiation gives us ) ) cos θ R sin θ θ γ T = sin θ s R cos θ s ; 71) solving for R/ s and θ/ s gives us R s = γ 1 cos θ γ sin θ, θ s = 1 7) R γ 1 sin θ γ cos θ). We note that R cannot be zero by assumption. Differentiating 67), we have f s = r µ γ s) R s = r µ γ s) γ 1 cos θ γ sin θ). 7) Combining 7), 7) and 68) gives the result. 7)... Stationary-phase analysis of F. If we carry out a stationary-phase reduction of 47) in the variables µ T and η T, we find ) π FVs,t) = e iφt,s,x,ω, µ T, µ T ) At, s, x,ω, µ T, µ T ) ω Vx T ) det D µ φ 1 s, x, µ T ) dω d x T E 1 s, t), 74) where µ T is shorthand for µ T s, x), and where the determinant of the Hessian of φ is given by det D φ s, x, µ T ) = det D µ φs,x, µ T ), 75)

16 5 M Cheney and C J Nolan with D µ φ [ s, x, µt ) = 1 H x c µ µ ) µ T /ω ) γ s) H) µ ) µ ) µt /ω )) H x γ ) ] s) H) µ µ, 76) and where E 1 denotes a function smoother than the other terms on the right-hand side. The details of the stationary-phase calculation are given in appendix B.... The canonical relation. The directional microlocal ) information about how F maps singularities is given by the twisted) canonical relation, which is a subset of the set of points s, t; σ, τ), x; ξ)) in phase space with σ = s φ,τ = t φ, and ξ = xt φ such that s, t, x,ω, µ T, η T ) is in the critical set 54) 56). Differentiating 48) and then using the fact that µ T = η T on the critical set, we find { = s, t; σ, τ), x; ξ)) essential support of A : t = γs) x H ɛ x H x )/c σ = s φ = 1 c [ µt,ω µ ) ηt,ω η )] γs) = kµ γs) 77) τ = t φ = ω ξ = xt φ = µ T η T c = kµ T Here the essential support of A denotes the subset of points on the critical set for which A does not correspond to an infinitely) smoothing operator i.e. A is not in the symbol class S = m S m ). In other words, for the purpose of studying how F maps singularities, points in the complement of the essential support of A can be neglected. In our case, we have removed troublesome points with our taper function ψ; points for which ψ is zero are clearly not in the essential support of A and thus are not in. }. 4. Image formation To form an image i.e., an approximation to V ), we would like to apply to the data an operator Q such that QF = I. Microlocal analysis shows us how to construct a relative parametrix Q for F so that QF = I relatively smoothing operator), where by a relatively smoothing operator we mean one that improves the smoothness. The operator Q is constructed as an FIO Qd)z) = e iφt,s,z,ω, µ T, η T ) Bt,s, z,ω, µ T, η T )ds, t) dω d η T d µ T ds dt, 78) whose phase is the negative of the phase of F. We note that the phase of 78) is the same as that of the adjoint F. At this point the amplitude B is not known; it will be determined below Composition of Q and F The composition of two FIOs does not always make sense. Here we rely on the following theorem [9], which gives a criterion involving the projection P from points s, t; σ, τ), x, ξ)) to the output variables s, t; σ, τ):

17 Synthetic-aperture imaging through a dispersive layer 5 Theorem 1. If the canonical relation of F is such that the projection P is an injective immersion i.e., is one-to-one and its derivative DP is also one-to-one), then F or Q) can be composed with F, and the composite operator F F resp. QF )isagainanfio. The conditions of this theorem are not satisfied unless we make an additional assumption about the antenna beam patterns. Assumption 8. The antenna beam pattern J s or J r directs radiation to one side of the flight path, so that the product J s ω, s, µ/ω) J r ω, s, η/ω) is negligible when [ µ s, x) γs)] and [ η s, x) γs)] the reverse inequalities hold if the radiation is directed to the other side). In other words, we assume that the essential support of A contains only points for which [γs) x H ) γs)] is strictly positive or strictly negative, depending on how the antenna is mounted). If this assumption is not satisfied, QF is not an FIO, and this implies that the image QF contains certain artefacts [8]. Proposition 5. Under assumptions 1 8, the canonical relation of F satisfies the hypotheses of theorem 1. The proof of this proposition is contained in appendix C. 4.. Determination of Q To determine the amplitude B that will make QF approximately the identity operator, we apply the operator Q to the data d = FV: QF V )z) = e iφt,s,z,ω, µ T, η T ) Bt,s, z,ω, µ T, η T ) e iφt,s,x,ω, µ T, η T ) At, s, x,ω, µ T, η T )V x T ) d µ T d η T dω d x T dω d η T d µ T ds dt, 79) where z = z T, ). We then carry out the following calculations Simplification of QF by stationary phase. In 79) we apply the method of stationary phase in the pair of variables ω,t. For this calculation, we write the phase φ as φt,s,x,ω, µ T, η T ) = ωt ϕs,x,ω, µ T, η T ), where µt ϕs,x,ω, µ T, η T ) = k ω γ T s) x T ) µ H x ) µ γ s) H) η ) T ω x T γ T s)) η H x ) η γ s) H) = k [ µ γs) x H ) µ x H x) η xh γs)) η x xh ) ], 8) where µ = µ T /ω, µ ) = µt,µ ), etc. A careful calculation requires that we use the homogeneity of ϕ and a number of changes of variables to bring out ω as the large parameter in order to apply the stationary-phase theorem; however, the end result is the same

18 54 M Cheney and C J Nolan as if we had simply used exp[itω ω )]dt = πδω ω ) and evaluated the result at t = t ϕ s, x,ω, µ T, η T ) = ϕs,x, 1, µ T /ω, η T /ω). Thus we obtain QF V )z T ) = π e iϕs,z,ω, µ T, η T ) Bt ϕ,s,z,ω, µ T, η T ) e iϕs,x,ω, µ T, η T ) At ϕ,s,x,ω, µ T, η T )V x T ) d µ T d η T d x T dω d η T d µ T ds E z T ), 81) where E is one degree smoother than the first term. Next we carry out a stationary-phase reduction in the variables µ T, η T, µ T, and η T. These calculations are the same as done in the stationary-phase analysis 74)ofF. This process results in QF V )z) = π)5 e i[ϕs,x,ω, µ T s,x), µ T s,x)) ϕs,z,ω, µ T s,z), µ T s,z))] ω 4 Bt ϕ,s,z,ω, µ T s, z), µ T s, z)) D µ φ 1 s, z, µ T s, z)) At ϕ,s,x,ω, µ T s, x), µ T s, x)) D µ φ 1 s, x, µ T s, x)) Vx T ) d x T dω ds E z), 8) where again E denotes a smoother term Change of variables. Next we expand the phase in a Taylor series about the point z T = x T : ϕs,x,ω, µ T s, x), µ T s, x)) ϕs,z,ω, µ T s, z), µ T s, z)) = z x) T z,s,x,ω), 8) where 1 z,s,x,ω)= yt ϕs,y,ω, µ T s, y), µ T s, y)) dλ. 84) y=xλz x) In 8) we now make the change of variables justified below) s, ω) ξ = z,s,x,ω), 85) which transforms 8) into QF V )z) = e iz x) T ξ Bt ϕ,s,z,ω, µ T s, z), µ T s, z)) At ϕ,s,x,ω, µ T s, x), µ T s, x)) D µ φ 1 s, z, µ T s, z)) D µ φ 1 s, x, µ T s, x)) s,ω) ξ z,s,x,ω)vx T ) d x T dξ E 4 z), 86) where in 86), s = sξ), ω = ωξ) and t ϕ = t ϕ s, x,ω, µ T, µ T ). We see that the phase of 86) is the same as that of a delta function δz x) exp[iz x) T ξ]dξ, which, together with the estimates of proposition, means that the operator QF is a pseudodifferential operator. Pseudodifferential operators have the pseudolocal property [44], i.e., they do not move singularities or change their orientation. This means that the imaging operator Q correctly reconstructs the edges and interfaces that are visible in the original scene. Mathematically, QF s preservation of visible singularities is due to the fact that the leading order contribution to 86) comes from the point z T = x T.

19 Synthetic-aperture imaging through a dispersive layer 55 Calculation of the Jacobian determinant. To calculate the Jacobian determinant when z = x, we note that since ϕs,x,ω, µ T s, x), µ T s, x)) = k γs) x H ɛ x H x ), 87) the ϕ appearing in 84)is xt ϕs,x,ω, µ T s, x), µ T s, x)) = k γs) x H ) D T x H ɛ xh x) D T x H I T )) = k[µ D T x H µ D T x H I T )], 88) where 1 I T = 1, 89) and D T x H is a matrix composed of the column vectors x D T x H H ) =, xh. x 1 x 9) We note that the bottom row of the matrices D T x H and I T implies that 88) can be written as is zero; since µ T = µ T,this ϕs,x,ω, µ T s, x), µ T s, x)) = kµ T D T x H D T x H I T ) = kµ T. 91) Thus at x = z, wehave ξ = kµ T. 9) This implies that the Jacobian determinant is ξ s,ω) = k µ s s µ 1 µ = 4k ) µ 1 µ c s µ µ 1. 9) s c c From corollary and 65), we find ξ s,ω) = 4k r γ µ r sin θ cos θ s) γ 1 cos θ γ sin θ) c γ s) H H x ɛ µ ) µ r R sin θ γ cos θ γ 1 sin θ) r γ µ r cos θ sin θ s) γ 1 cos θ γ sin θ) γ s) H H x ɛ µ ) µ r R cos θ γ cos θ γ 1 sin θ) [ ] = 4k r c R γ cos θ γ 1 sin θ) = 4k r c R γ T s) µ T, 94) where µ T = rsin θ, cos θ).

20 56 M Cheney and C J Nolan Explicit calculation of this expression for use in the backprojection operator requires knowledge of µ T in terms of s and x. We see from 94) that the change of variables 85) can be made everywhere except along the tangent to the flight path. This region is already excluded by assumption The imaging operator. We see from 87) that we can use a somewhat simplified form of Q, namely Qd)z) = e iωt [ γs) zh ɛ z H z ]/c ) Bz,s,ω)ds,t)dωds dt, 95) where ξ s,ω) z,s,z,ω) D µ φ s, z, γs) zh ) T )χz,s,ω) Bz,s,ω)=, 96) Aϕs, z, 1, γs) zh, zh γs)), s, z,ω, γs) zh, zh γs)) where χz,s,ω)is a smooth cutoff function that prevents division by zero in 96). Stationary phase analysis of QF shows, as above, that QF = I smoothing operator of degree 1); this means that the operator Q, when applied to the data, produces a bandlimited version of an image in which the singularities of V have the correct positions and strengths. In particular, the leading order behaviour of the kernel of QF is the approximate delta function exp[iz x)t ξ]dξ. The degree to which the point spread function kernel of QF ) approximates a delta function i.e. the resolution of the system) is determined by the region of ξ integration. This region is determined, via relation 9), by the bandwidth, antenna beam pattern and geometry of the data collection curve. The resolution is actually slightly better than if the dispersive layer were absent; this is because when the ray path from γs) to x is bent downwards by the refracting medium as in figure ), the horizontal components of µ T are slightly larger than they would be if the ray path were straight. The image of the singularities of V could be enhanced by using, instead of Q, an operator Q 1, which is the same as Q except that B is replaced by B 1 = ξ B = k µ T B. The leading order behaviour of Q 1 F is then ξ exp[iz x) T ξ]dξ, a pseudodifferential operator of order 1. The resulting image would be an approximation to V, in which jump discontinuities of V would appear as bandlimited) delta functions. It is theoretically possible to obtain an approximation Q # to F 1 that, rather than being good only to first order as are Q and Q, is good to all orders of smoothness. This could be done by using the full stationary-phase expansion [19] in the places where we have used simply the first-order stationary-phase approximation. The amplitude of the resulting operator Q # would be more complicated, but its leading order term would be given by 96). For dispersive materials, ɛ r ω) is always complex, which implies that such materials are dissipative to some degree. This, in turn, implies that for some problems the amplitude A may be quite small, and consequently the inversion of 96) may tend to amplify noise. Investigation of this issue is left for the future. 5. Conclusions We have shown an image formation operator 95) that can be used for synthetic-aperture imaging through a homogeneous dispersive layer. We see from the phase of 95) that it is the refraction through the layer that must be accounted for in order to obtain a focused image. The dispersion can be corrected for by adjusting the amplitude B of the backprojection operator 95). This amplitude, however, does not affect the positioning of the edges in the image.

21 Synthetic-aperture imaging through a dispersive layer 57 The refraction through the layer is determined by the high-frequency permittivity ɛ. We leave for the future the following questions: a) Can the high-frequency quantity ɛ be determined [] from bandlimited data? b) If ɛ is estimated incorrectly, what effect does this have on the image? c) What is the best way to implement the algorithm numerically? and d) How does the algorithm behave in the presence of noise? Acknowledgments We are grateful to Richard Albanese and Arje Nachman for drawing attention to the issue of dispersion in foliage-penetrating SAR. We thank Tom Roberts for helpful discussions regarding dispersive materials, and Sherwood Samn for reading the manuscript and catching a number of minor errors. MC s work was sponsored by the Air Force Office of Scientific Research under agreement number F This work was also supported in part by the National Science Foundation through its Engineering Research Centers Program award number EEC ) and its Focused Research Groups in the Mathematical Sciences program. Appendix A. Estimates for the amplitude ã We recall that ã is defined by ãω, η T ) = aω, η T /ω) = ψω, η T /ω) e ik η aω, η T /ω)h z ). We consider the various components of ã separately. A.1) Lemma 1. If assumption 6 holds, and in addition η T is strictly bounded away from ɛ, then η a is smooth, and we have β 1 η 1 β η β ω η aω, η T ) Cβ 1ω ) 1β)/, A.) where β 1,β and β are non-negative integers. The constant C β is zero if β 1 > or β > or β 1 β. Proof. When β =, we have η a ω, η T ) = η ω, η T ) η η T ) = ɛ ɛ s ω) η T ɛ η T = ) ɛ ɛ η T 1 s ω) ɛ η T ) 1 [ = η η 1 ɛ s ω) T ) η ) O ɛ s ω) )] η ) ) = 1 ɛ s ω) ɛs ω) η O η ) = O1ω ) 1/ ), A.) Consequently the US Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Research Laboratory or the US Government.

22 58 M Cheney and C J Nolan Table 1. The type of terms obtained from differentiation of ζ. Derivative Contains terms of the form ω ζ ζ 1/ ω ζ) ηj ζ ζ 1/ ηj ζ ) ηj ω ζ ζ / ηj ζ ) ω ζ), ζ 1/ ηj ωζ = η j ω ζ ζ 5/ ηj ζ ) ω ζ), ζ / η j ζ ) ω ζ) ω ζ ζ / ω ζ), ζ 1/ ω ζ ηj ω ζ ζ 5/ ηj ζ ) ω ζ), ζ / ηj ζ ) ω ζ ) Table. The type of terms obtained from differentiation of h = ωη a. Derivative h = ωη a ω h ηj h ω h ηj ω nh Contains terms of the form ωη a η a, ω ω η a ω ηj η a ωη a ), ω ω η ) a η j ω n 1 η a, ω ηj ω nη a where in the fourth line we have used the expansion 1x = 1x/Ox ). We see that the estimate A.) is satisfied away from η T = ɛ where η is zero). To check the derivatives, we use the temporary notation η a = ζ ζ, where ζω,η) = ɛ ɛ s ω) η T and ζ η) = ɛ η T. We note that ω ζ = ω ɛ s, ω ζ =, ηj ζ = ηj ζ = η j, all mixed derivatives of ζ and ζ are zero and all η-derivatives of order higher than are zero. Derivatives of ζ are shown in table 1. Differentiations with respect to η j are bounded because η j is bounded. The reciprocal powers of ζ are bounded because the microlocal cutoff ψ causes η T to be strictly bounded away from ɛ ɛ s ω). We see that it is the order of the derivative with respect to ω that determines the large-ω behaviour. The only nonzero derivatives of ζ are those with respect to η j ; they follow the same pattern as the derivatives of ζ. We write h = ωη a. Lemma. If assumption 6 holds, and in addition η T is strictly bounded away from ɛ, then h is smooth, and we have β 1 η 1 β η β ω hω, η T ) Cβ 1ω ) β/, A.4) where β 1,β and β are non-negative integers. The constant C β is zero if β 1 > or β > or β 1 β. Proof. We show the derivatives of h in table. To the terms arising from ω n h, we apply lemma 1. Proof of proposition 1. We note that from the chain rule, we can write ω ã = ω a η T /ω) ηt a. Since η T is bounded, and further differentiations cannot introduce positive powers of ω, we see that 51) will be proved if we prove the version of 51) with the tildes removed.

23 Synthetic-aperture imaging through a dispersive layer 59 Table. The type of terms obtained from differentiation of a. Derivative Contains terms of the form a multiplied by a = ψ e h 1 ω a ω h ηj a ηj h ω a ωh), ω h) η j a ηj h ), ηj h ) ηj ω a ηj h ) ω h), ηj ω h ), ω a ωh), ω h) ω h), ω h ) η j ω a ηj h ) ω h), η j h ) ω h), ηj h ) ηj ωh ), η j ω h ηj ω a ηj ω h ) ω h), ηj h ) ω h), ηj h ) ω h), ηj ω h η j a ηj h ), ηj h ) η j h ), η j h = ω na ωh) n,..., ω nh We show derivatives of a = ψ e h in table. Here we ignore terms involving derivatives of ψ because ψ is chosen so that all its derivatives decay more rapidly than any polynomial. We ignore also factors of H z)/ c. Combining table with the results of lemma shows that the large-ω decay corresponds to the number of derivatives with respect to ω; this gives proposition 1. Appendix B. Stationary-phase calculations The stationary-phase theorem states that [4, 19, ] Theorem. If a is a smooth function of compact support on R n, and φ has only non-degenerate critical points, then as ω, ) π n/ e iωφx) ax) d n x = ax ) eiωφx) e iπ/4)sgnd φx ) Oω n/ 1 ). ω {x :Dφx )=} det D φx ) B.1) Here Dφ denotes the gradient of φ, D φ denotes the Hessian, and sgn denotes the signature of a matrix, i.e., the number of positive eigenvalues minus the number of negative ones. B.1. Stationary-phase analysis of 47) To obtain 74), we need the Hessian D D ) φ = µ φ Dη φ, B.) where ) Dµ φ = a11 a 1, B.) a 1 a with the a computed from 55)tobe [ a ii = 1 H ) x 1 µ ) i 1 µ i γ ωc µ ω µ ) s) H) µ ω ) µ )], ) a 1 = µ 1 µ x H γ B.4) s) H ω ) c µ, µ