Wave propagation and sampling theory—Part II: Sampling theory and complex waves

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Morlet et al (1982, this issue) showed the advantages of using complex values for both waves and characteristics of the media. We simulated the theoretical tools we present here, using the Goupillaud‐Kunetz algorithm. Now we present sampling methods for complex signals or traces corresponding to received waves, and sampling methods for complex characterization of multilayered or heterogeneous media. Regarding the complex signals, we present a two‐dimensional (2-D) method of sampling in the time‐frequency domain using a special or “extended” Gabor expansion on a set of basic wavelets adapted to phase preservation. Such a 2-D expansion permits us to handle in a proper manner instantaneous frequency spectra. We show the differences between “wavelet resolution” and “sampling grid resolution.” We also show the importance of phase preservation in high‐resolution seismic. Regarding the media, we show how analytical studies of wave propagation in periodic structured layers could help when trying to characterize the physical properties of the layers and their large scale granularity as a result of complex deconvolution. Analytical studies of wave propagation in periodic structures are well known in solid state physics, and lead to the so‐called “Bloch waves.” The introduction of complex waves leads to replacing the classical wave equation by a Schrödinger equation. Finally, we show that complex wave equations, Gabor expansion, and Bloch waves are three different ways of introducing the tools of quantum mechanics in high‐resolution seismic (Gabor, 1946; Kittel, 1976; Morlet, 1975). And conversely, the Goupillaud‐Kunetz algorithm and an extended Gabor expansion may be of some use in solid state physics.