Principles of Modern Radar - Volume 2 1891121537

5.4 Sample Radar Applications 189tunnels are simple and have exploitable structure, such as well-defined sharp edges and lowwithin-target contrast. In this example we will show how a priori knowledge of structuredsparsity in the data allows us to appropriately regularize the otherwise ill-posed problemof underground imaging from an array of ultra narrowband antennas.In all of our previous examples, we have assumed free-space propagation between theradar antennas and the scattering object while solving for a nondispersive target reflectivityfunction, V (q) = 1 − ɛ(q) (this formulation necessarily assumes that the scatteringɛ 0material is isotropic and the materials have constant magnetic permeability μ r that can beabsorbed into the scattering reflectivity). In the case of GPR, the situation is complicatedby the air–ground interface and possible frequency-dependent attenuation through theground. In the context of the Born approximation, Deming [169] provides an algorithm thatreplaces the free-space Green’s function with one that more accurately models propagationthrough lossy earth. Various formulations of the Green’s function that also include the air–soil interface effects can be found in [170–173].Assuming the Born approximation is reasonable and the direct-path contribution tothe data has been removed, a generic linear model for the above-ground bistatic measuredelectric field (using dipole antennas of length l) scattered from isotropic, nonmagnetic,homogeneous soil is [174]∫ ∫ ∫E scatt (γ t , γ r ,ω)= iωμ 0 k0 2 l br T G(γ r, q,ω)· G(q, γ t ,ω)a} {{ } t V (q)d 3 q (5.46)E incwhere we have introduced the scalar k 0 = ω/c 0 , the 3 × 3 dyadic Green’s functionG, the transmit and receive locations γ t , γ r ∈ R 3 , and the vectors a t , b r ∈ C 3 thatencode the orientation, gain, and phase of the transmit and receive dipoles, respectively.If∑P transmitters simultaneously transmit, then the incident field is modeled as E inc =Pp=1 G(x′ , γ tp ,ω)a tp , and the measured field (5.46) at the receiver becomes⎛⎞∫ ∫ ∫P∑E scatt (γ r ,ω)= iωμ 0 k0 2 l br T G(γ r, q,ω)· ⎝ G(q, γ tp ,ω)a tp⎠ V (q)d 3 q(5.47)Equation (5.47) represents a linear equation that can be discretized to solve for the imageV . By unwrapping the 3-D image V into the vector x ∈ C N , concatenating the measurementsof the scattered field E scatt (γ r ,ω)into a vector y ∈ C M with M =(number oftransmit frequencies)(number of transmitter/receiver pairs), and discretizing the forwardoperatorA ∈ C M×N into elements A m,n whose row number m indexes over all of thetransmitter/receiver/frequency combinations and column number n indexes the voxels inthe scene, we arrive at our usual ill-posed formulation Ax = y.For our numerical example consider a synthetic transmitter/receiver setup in which thetransmitters and receivers are arranged in a circular array placed on the air-soil interface30 meters above an L-shaped tunnel (see Figure 5-12 for a top-down view of the sensorconfiguration and target). The forward data were simulated with the FDTD simulatorGPRMAX [175] over the 4-7 MHz frequency range with 250 KHz steps. Although nonoise was intentionally added to the data, the effects from invoking the Born approximationin the inversion algorithm, discretization errors, and imperfect removal of the direct-pathsignal were included. The number of measurements is M = 360, and the number of pixelsis N = 8405.p=1