Scattering and; Delay, Scale, and Sum Migration

Abstract

How do we see? What is the mechanism? Consider standing in an open field on a clear sunny day. In the field are a yellow dog and a blue ball. From a wave-based remote sensing point of view the sun is a source of radiation. It is a broadband electromagnetic source which, for the purposes of this introduction, only the visible spectrum is considered (approximately 390 to 750 nanometers or 400 to 769 TeraHertz). The source emits an incident field into the known background environment which, for this example, is free space. The incident field propagates until it strikes an object or target, either the yellow dog or the blue ball. The interaction of the incident field with an object results in a scattered field. The scattered field arises from a mis-match between the background refractive index, considered to be unity, and the scattering object refractive index ('yellow' for the case of the dog, and 'blue' for the ball). This is also known as an impedance mis-match. The scattering objects are referred to as secondary sources of radiation, that radiation being the scattered field which propagates until it is measured by the two receivers known as 'eyes'. The eyes focus the measured scattered field to form images which are processed by the 'wetware' of the brain for detection, identification, and localization. When time series representations of the measured scattered field are available, the image forming focusing process can be mathematically modeled by delayed, scaled, and summed migration. This concept of optical propagation, scattering, and focusing have one-to-one equivalents in the acoustic realm. This document is intended to present the basic concepts of scalar scattering and migration used in wide band wave-based remote sensing and imaging. The terms beamforming and (delayed, scaled, and summed) migration are used interchangeably but are to be distinguished from the narrow band (frequency domain) beamforming to determine the direction of arrival of a signal, and seismic migration in which wide band time series are shifted but not to form images per se. Section 3 presents a mostly graphically-based motivation and summary of delay, scale, and sum beamforming. The model for incident field propagation in free space is derived in Section 4 under specific assumptions. General object scattering is derived in Section 5 and simplified under the Born approximation in Section 6. The model of this section serves as the basis in the derivation of time-domain migration. The Foldy-Lax, full point scatterer scattering, method is derived in Section 7. With the previous forward models in hand, delay, scale, and sum beamforming is derived in Section 8. Finally, proof-of-principle experiments are present in Section 9.