Imaging and Inversion with Acoustic and Elastic Waves

Imaging the interior of a region with waveforms recorded on the surface is a key application of scattering theory in geophysics, medical imaging and many other fields. In this chapter I will discuss how the waveform inversion problem can be formulated in terms of statistical inference. Statistical methods are essential in order to be able to take into account the limited, uncertain nature of the observations. The goal will not be to find the “true” model of the sampled region (i.e., its boundaries and/or material properties), since there will always be infinitely many models that fit the data, but rather to put bounds on the feasible values of the parameters that characterize the model. While the idea of using inference methods in waveform inversion is not new, as proposals along these lines were made at least as far back as the mid-1980s, field data applications with realistic prior information and uncertainty estimates are rare because of the difficulties in quantifying this information. In addition, it will also be seen that the classical imaging methods of seismology and ultrasonics (such as migration and diffraction tomography) arise as special cases of more general waveform inversion formulae. After giving a general formulation of the waveform inversion problem, I will show an example of the application of these ideas to elastic waveforms from a reflection seismic experiment. While the application shown is drawn from exploration geophysics, the methods described are applicable in many fields that make use of arraybased waveform measurements, such as ultrasonics, non-destructive testing, global seismology, ocean acoustics and ground-penetrating radar. 0.1.1. Seismic Imaging