Radar Imaging with Independently Moving Transmitters and Receivers

In this paper we present an analytic filtered-backpropagation inversion method for radar imaging when the transmitters and receivers are moving along independent flight paths. The flight paths are arbitrary, but assumed known. We assume a singlescattering model for the radar data, and we assume that the ground topography is known. We assume, in addition, that the receiver can correctly associate the data with the right transmitter. I. I NTRODUCTION For a long time there has been interest in radar imaging in the case when the transmitters and receivers are not co-located ([1], [7], [12], [13], [14], [15]). Much of this previous work deals with particularly simple trajectories such as lines and circles. In this paper we show how arbitrary trajectories can be handled. Our approach is first to develop a physics-based model for the received signal (section II), and then to develop an approximate inversion method (section III). Our approximate inversion method is based on the microlocal-analysis techniques that were first introduced for seismic imaging ([3], [2], [5]) and were recently extended to radar imaging ([9], [10], [6], [16], [11]). These techniques give rise to filtered-backpropagation algorithms, and the resulting images have the desirable property that visible edges in the scene appear in the correct location with the correct orientation. II. T HE MATHEMATICAL MODEL We develop the mathematical model in the frame of reference in which the target is not moving, but both transmitter and receiver may be moving. Fig. 1. Geometry A. A model for the wave propagation A scalar approximation to Maxwell’s equations is ∇E(t, x)− (μ0ε(x)Ë(t, x)) = −μ0j̇(t, x). (1) where the dots denote differentiation with respect to time t, μ0 is the magnetic permeability of free space, ε is the electric permittivity, andj is proportional to the effective current density that is the source of the field. We writeμ0ε(x) = μ0ε0+ρ(x) = c−2 0 +ρ(x); this converts (1) into ∇E(t, x)−c−2 0 ∂ t E(t, x) = ρ(x)Ë(t, x)−μ0j̇(t, x), (2) B. A linearized scattering model The field incident on the reflectivity distribution ρ is the free-space fieldE radiating from the antenna: ( ∇ − c−2 0 ∂ t ) E(t, x) = −μ0j̇(t, x) (3) We writeE = E+E in (2) and use (3) to obtain ( ∇ − c−2 0 ∂ t ) E(t, x) = ρ(x)Ë(t, x). (4) We can write (4) as an integral equation E(t, x) = − ∫ g(t− t′,x− z)ρ(z)Ë(t′,z)dt′dz, (5) where the outgoing Green’s function for the wave equation is g(t, x) ∝ δ(t− |x|/c0) 2|x| = ∫ e−iω(t−|x|/c0) 4π|x| dω (6) A commonly used approximation [8], often called the Born approximationor the single scattering approximation, is to replace the full fieldE on the right side of (5) and (4) by the incident field E, which converts (5) to E B (t, x) ≈ − ∫ g(t− t′,x− z)ρ(z)Ëin(t′,z)dt′dz (7) The value of this approximation is that it removes the nonlinearity in the inverse problem: it replaces the product of two unknowns ( ρ andE) by a single unknown (ρ) multiplied by the known incident field. C. A model for the field from an antenna The source of the field radiated from the antenna is the effective current density j on the antenna. The radiated field is the solution of (3), namely E(t, x) = ∫ g(t− t′,x− y)μ0j̇(t,y)dt dy. (8) We note that causality is evident from (8) and (6): E(t, x) is affected only by those parts of the current densitẏj(t′,y) for which t−t′ = |x−y|/c0. We write −μ0j̇(t,y) in terms of its Fourier transformJ: μ0j̇(t,y) = ∫ e−iωt ′ J(ω, x)dω, (9) whereω denotes the angular frequency. With the notation (9) and (6), (8) becomes E(t, x) = ∫ e−iω(t−|x−y|/c0) 4π|x− y| J(ω, y)dωdy. (10) Expression (10) has the advantage of showing clearly the bandlimited nature of E. Next we assume that the antenna is small compared with the distance to the scatterers. We denote the center of the antenna by y; thus a point on the antenna can be written y = y + y′, wherey′ is a vector from the center of the antenna to the point y on the antenna. In this notation, the assumption that the scattering locationx is far from the antenna can be expressed|y′| << |x− y|. For suchx, we can write |x−y| = |x−y|−(x̂− y0)·y′+O(|y′|2/|x−y0|), (11) whereŷ denotes a unit vector in the same direction asy. We use the expansion (11) in (10) to obtain E(t, x,y) ≈ ∫ e−iω(t−|x−y |/c0) 4π|x− y0| e−iω(x̂−y 0)·y′ Js(ω, y + y′)dωdy′ ≈ ∫ e−iω(t−|x−y |/c0) 4π|x− y0| Ĵ(ω, x,y)dω