K. A. Innanen, Methods for the treatment of acoustic and absorptive/dispersive wave field measurements, 2003.
Many recent methods of seismic wave field processing and inversion concern themselves with the fine detail of the amplitude and phase characteristics of measured events. Processes of absorption and dispersion have a strong impact on both; the impact is particularly deleterious to the effective resolution of images created from the data. There is a need to understand the dissipation of seismic wave energy as it affects such methods. I identify: algorithms based on the inverse scattering series, algorithms based on multiresolution analysis, and algorithms based on the estimation of the order of the singularities of seismic data, as requiring this kind of study. As it turns out, these approaches may be cast such that they deal directly with issues of attenuation, to the point where they can be seen as tools for viscoacoustic forward modelling, Q estimation, viscoacoustic inversion, and/or Q compensation.
In this thesis I demonstrate these ideas in turn. The forward scattering series is formulated such that a viscoacoustic wave field is represented as an expansion about an acoustic reference; analysis of the convergence properties and scattering diagrams are carried out, and it is shown that (i) the attenuated wave field may be generated by the nonlinear interplay of acoustic reference fields, and (ii) the cumulative effect of certain scattering types is responsible for macroscopic wave field properties; also, the basic form of the absorptive/dispersive inversion problem is predicted. Following this, the impact of Q on measurements of the local regularity of a seismic trace, via Lipschitz exponents, is discussed, with the aim of using these exponents as a means to estimate local Q values. The problem of inverse scattering based imaging and inversion is treated next: I present a simple, computable form for the simultaneous imaging and wavespeed inversion of 1D acoustic wave field data. This method is applied to 1D, normal incidence synthetic data; its sensitivity with respect to contrast, complexity, noise and bandlimited data are concurrently surveyed. I next develop and test a Born inversion for simultaneous contrasts in wavespeed and Q, distinguishing between the results of a pure Born inversion and a further, bootstrap, approach that improves the quality of the linear results. The nonlinear inversion subseries of the inverse scattering series is then cast for simplified viscoacoustic media, to understand the behaviour and implied capabilities of the series/subseries to handle Q. The ``communication between events'' of the inversion subseries is developed in theory and with numeric examples; it is shown that terms which contain cumulative information from all portions of the data dominate over local terms in determining correct, local, model amplitudes. Finally, I consider the use of a wavelet-based regularization of the operator for viscoacoustic downward continuation.
The inclusion of absorption and dispersion in the theory that underlies many seismic methods leads to processing and inversion methods that estimate attenuation parameters and compensate for unwanted effects. These methods are sensitive to amplitude and phase information (by design) and so require low noise, often broadband data; however the methods have responded very favourably to synthetic data tests, and tend to be forgiving to bandlimited data with small amounts of error.