The Dallas Geophysicical Society is very grateful to Carlos Moreno, President of Lumina, for graciously stepping in to give the presentation, when Dr. Castagna was unable to come.  Biographical information may be found by clicking on his name above.

Spectral inversion is a term that was coined by Greg Partyka in his 2005 SEG Distinguished lecture tour and is a logical outgrowth of spectral decomposition interpretation when the local seismic response is not dominated by a single layer. The idea of spectral inversion is that the local frequency response of a seismic signal resulting from spectral decomposition is a function of the local reflectivity spectrum, and thereby, is an avenue of inversion for reflection coefficients. In the simplest kind of spectral inversion, we assume that local frequency spectra are a superposition of a limited number of layer responses, each of which are sinusoids in the frequency domain tapered by the wavelet spectrum (Puryear and Castagna, 2008). Although it is true that any operation in the frequency domain has an equivalent operation in the time domain, as implemented, spectral inversion results in a different output than conventional time-domain algorithms, such as sparse-spike inversion, that similarly do not require a starting model. This is because, whereas sparse-spike algorithms are designed to control the sparsity of individual reflection coefficients in a series, spectral inversion is based on the assumption that a limited number of layers dominate the seismic response, and thus, the method does not discriminate against seismically thin beds if their response is of sufficient amplitude to contribute significantly to the local spectrum in the presence of noise.

All sparse inversion algorithms, be they implemented in the time or frequency domain, result in output reflectivity series that have frequency content outside the band of the original data. This is entirely a consequence of the assumption that the earth is primarily blocky, which happens to be a good assumption for most sedimentary rocks. The assumption of blockiness is a form of a priori information, which can be shown theoretically to produce earth models with greater frequency content than the original data. This is not surprising or new to experts in seismic inversion theory. Arguments about the impossibility of restoring lost frequencies outside the seismic band using deconvolution, though correct in the context of inverse digital filtering, are simply incorrect and not applicable to seismic inversion in the presence of a priori information. The real issue is not whether or not valid frequency information outside the seismic band can be obtained, but rather, how useful is this necessarily incomplete additional frequency content in creating an image, and how robust is the resulting image when the earth impedance structure is transitional or completely random rather than blocky.   Our investigations, including many synthetic and field examples, suggest that the method is usually not only robust, but that the additional frequency content does indeed improve temporal and spatial resolution.    We have found that, when the assumptions of the method are applicable and the wavelet is well known, layer resolution below 1/8^th a wavelength can be achieved, with the limiting factor being the S/N ratio of the original data. We have also found that the broader the input seismic bandwidth, the greater the immunity to noise and the greater the resolution improvement that can be achieved.