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UBC Theses and Dissertations

Complex seismic data : aspects of processing, analysis, and inversion Scheuer, Timothy Ellis


A common theme in this thesis is the use of complex signal attributes to facilitate the processing, analysis, and inversion of seismic data. Complex data are formed from real data by removing the negative frequency portion of the Fourier transform and doubling the positive frequency portion. Removing the dual nature of frequency components in real data allows the development of efficient algorithms for time-variant filtering, local phase velocity estimation, and subsequent velocity-depth inversion of shot gathers using Snell traces. For 1-D seismic data, I develop a computationally efficient time-variant filter with an idealized boxcar-like frequency spectrum. The filter can either be zero phase, or it can effect a time-variant phase rotation of the input data. The instantaneous low-frequency and high-frequency cutoffs are independent continuous time functions that are specified by the user. Frequency-domain rolloff characteristics of the filter can be prescribed, but the achieved spectrum depends on the length of the applied filter and the instantaneous frequency cutoffs used. The primary derivation of this theory depends upon the properties of the complex signal and the complex delta function. This formulation is particularly insightful because of the geometrical interpretation it offers in the frequency domain. Basically, a high-pass filter, can be implemented by shifting the Fourier transform of the complex signal towards the negative frequency band, annihilating that portion of the signal that lies to the left of the origin, and then shifting the truncated spectrum back to the right. This geometrical insight permits inference of the mathematical form of a general time-variant band-pass filter. In addition, I show that the time-variant filter reduces to a Hilbert transform filter when the derivation is constrained to include real signal input. Application of the procedure to a spectral function permits frequency-variant windowing of an input time signal. For 2-D arid 3-D seismic data, I propose a new method that uses the concepts of complex trace analysis for the automatic estimation of local phase velocity. A complex seismic record is obtained from a real seismic record by extending complex trace analysis into higher dimensions. Phase velocities are estimated from the complex data by finding trajectories of constant phase. In 2-D, phase velocity calculation reduces to a ratio of instantaneous frequency and wavenumber, and thus provides a measure of the dominant plane-wave component at each point in the seismic record. The algorithm is simple to implement and computational requirements are small; this is partly due to a new method for computing instantaneous frequency and wavenumber which greatly simplifies these calculations for 2-D and 3-D complex records. In addition, this approach has the advantage that no a priori velocity input is needed; however, optimum stability is achieved when a limited range of dipping events is considered. Preconditioning the record with an appropriate velocity filter helps reduce the detrimental effects of crossing events, spatial aliasing, and random noise contamination. Accurate recovery of local phase velocity information about underlying seismic events allows the rapid evaluation of seismic attributes such as rms velocity and maximum depth of ray penetration. I utilize local phase velocity data from a shot gather for the estimation and inversion of Snell traces. The primary Snell trace corresponding to a 1-D velocity model locates all primary reflection energy,corresponding to a fixed emergence angle. Constraints on interval velocity and thickness obtained from several estimated Snell trajectories are inverted using SVD to provide a least squares velocity-depth model. The estimation and inversion is efficiently carried out on an interactive workstation utilizing constraints from a hyperbolic velocity analysis. Finally, Snell trace inversion is extended to an inhomogeneous medium. When dips are small, averaging Snell traces of a common phase velocity from forward and reversed shot gathers approximately removes the effects of planar dip. This allows recovery of velocity and depth vertically beneath the midpoint of the source locations used to obtain the reversed information.

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