Clifford J. Nolan

Synthetic aperture inversion

This paper considers synthetic aperture radar and other synthetic aperture imaging systems in which a backscattered wave is measured from a variety of locations. The paper begins with a (linearized) mathematical model, based on the wave equation, that includes the effects of limited bandwidth and the antenna beam pattern. The model includes antennas with poor directionality, such as are needed in the problem of foliage-penetrating radar, and can also accommodate other effects such as antenna motion and steering. For this mathematical model, we use the tools of microlocal analysis to develop and analyse a three-dimensional imaging algorithm that applies to measurements made on a two-dimensional surface. The analysis shows that simple backprojection should result in an image of the singularities in the scattering region. This image can be improved by following the backprojection with a spatially variable filter that includes not only the antenna beam pattern and source waveform but also a certain geometrical scaling factor called the Beylkin determinant. Moreover, we show how to combine the backprojection and filtering in one step. The resulting algorithm places singularities in the correct locations, with the correct orientations and strengths. The algorithm is analysed to determine which information about the scattering region is reconstructed and to determine the resolution. We introduce a notion of directional resolution to treat the reconstruction of walls and other directional elements. We also determine the fineness with which the data must be sampled in order for the theoretical analysis to apply. Finally, we relate the present analysis to previous work and discuss briefly implications for the case of a single flight track.

Synthetic aperture inversion for arbitrary flight paths and nonflat topography

This paper considers synthetic aperture radar (SAR) and other synthetic aperture imaging systems in which a backscattered wave is measured from positions along an arbitrary (known) flight path. The received backscattered signals are used to produce an image of the terrain. We assume a single-scattering model for the radar data, and we assume that the ground topography is known but not necessarily flat. We focus on cases in which the antenna footprint is so large that the standard narrow-beam algorithms are not useful. We show that certain artifacts can be avoided if the antenna and antenna footprint avoid particular relationships with the ground topography. We give an explicit backprojection imaging algorithm that corrects for the ground topography, flight path, antenna beam pattern, source waveform, and other geometrical factors. For the case of a non-directional antenna, the image produced by the above algorithm contains artifacts. For this case, we analyze the strength of the artifacts relative to the strength of the true image. The analysis shows that the artifacts can be somewhat suppressed by increasing the frequency, integration time, and the curvature of the flight path.

Enhanced angular resolution from multiply scattered waves

Multiply scattered waves are often neglected in imaging methods, largely because of the inability of standard algorithms to deal with the associated non-linear models. This paper shows that by incorporating a known environment into the background model, we not only retain the benefits of imaging techniques based on linear models, but also obtain different views of the target scatterer. The net result is an enhanced angular resolution of the target to be imaged. We carry out our analysis in the context of high-frequency radar imaging, in which a steerable beam from a moving platform is used to produce an image of a region on the earths surface (the target scatterers being buildings, etc). We consider the case where the target we want to image is situated in the vicinity of an a priori known reflecting wall. This is one of the simplest possible multipathing environments for the scatterer, and in the case when the illuminating beam is narrow enough to isolate different scattering paths, we will show that the imaging process achieves enhanced angular resolution. Although we carry out our analysis here in the context of radar, our technique is general enough that it can be adapted to many imaging modalities, such as acoustics, ultrasound, elasticity, etc. The extension of the method to other more complicated environments is also possible.

Imaging and coherency in complex structures

In complex structures, more than one pair of rays may connect a source and receiver via a scattering point. Whenever many such pairs of rays exist and have the same two-way traveltime, an inherent ambiguity appears in the inverse scattering process. Unless a simple ray geometric condition is met, the migrated section will contain images that need not belong to the actual reflectivity field. The geometric requirement relates to the acquisition geometry, and it is almost always satisfied when the sources and receivers both cover an area. Even if the geometric requirment is satisfied for the acquisition geometry, it may not be so for single source experiments (different acquisition geometry) consequently coherency panels may not be flat, even if one migrates with the exact background velocity.

Scattering in the Presence of Fold Caustics

We consider the linearized scattering operator

Synthetic-aperture imaging through a dispersive layer

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Anomalous Reflections Near a Caustic

of acoustics which maps singularities in the sound speed in the subsurface of the earth to singularities in the measured pressure at the earths surface. The source of the acoustic signal is concentrated at a point on the surface. Application ofthe adjoint scattering operator

Imaging from multiply scattered waves

{\cal F}^*

Microlocal analysis of synthetic aperture radar imaging in the presence of a vertical wall

to the measured pressure is a standard technique of locating sound speed singularities. The resulting image is equivalent to application of

MICROLOCAL ANALYSIS OF SAR IMAGING OF A DYNAMIC REFLECTIVITY FUNCTION

{\cal F}^*{\cal F}