ABSTRACT
As a wavefi eld propagates through a medium, it is subjected to time delays and loss of power. The wavefi eld is refl ected from interfaces separating media of different impedances, and is scattered by inhomogeneities present in the medium. By observing these effects, it is possible to study the characteristics of the medium through which the fi eld has propagated. Seismic exploration, on which depends the future discoveries of the petroleum deposits, exploits these effects of propagation to produce a detailed image of the subsurface geologic structure, which may be conducive to the accumulation of the hydrocarbon deposits. Likewise, the ultrasonic imaging used in medical diagnoses and in nondestructive testing also exploits the propagation effects of the wavefi eld. In this chapter, we shall study these effects of propagation for the purpose of constructing a three-dimensional image of the medium. Tomography refers to cross-sectional imaging of objects from transmitted, refl ected, or diffracted wavefi elds. Accordingly, there are three different types of tomographic-imaging methods. One or more effects of propagation, such as accumulated attenuation, travel time, wavefi eld produced by diffraction or scattering, are observed in all directions (360° for 3D imaging). The observations such as travel time delays or accumulated attenuation are inverted by solving a system of linear equations. Where the medium is a diffracting type, that is, the size of inhomogeneities is comparable to the wavelength of the illuminating wavefi eld, the preferred approach is Fourier inversion. The subject of tomographic imaging is covered in the next four sections. In the last section, we investigate how to estimate the shape of an object from its scattered fi eld.
