All practical wave equations are derived with common assumptions and simplifications in order to make their solution tractable. In this dissertation I will explore the concept of the "high-frequency" approximation, and describe several ways in which I have attempted to extend the range of validity down into the lower frequencies which are most commonly found in seismic exploration geophysics.
I have done this in several ways. First, I have shown that the eikonal equation may be extended to give useful results in lower frequencies by simply smoothing the underlying wavespeed velocity model of the medium in a frequency-dependent fashion. Second, I have shown that a similar kind of frequency-dependent smoothing may also be applied to the design of the Generalized Phase-Shift plus Interpolation (GPSPI) algorithm wavefield extrapolation operator, and this too yields higher fidelity in the extrapolation, especially at lower frequencies. Additionally, I have taken theoretical mathematical extensions to this same operator, and developed them into a practical and useful operator. In another study, I have shown that the Early Arrival Waveform Tomography method, a low-frequency extension of more common traveltime tomography, may be feasible to use for the time-lapse monitoring of changing petroleum reservoirs. Following this, I show that planewave imaging can dramatically reduce imaging computation time by introducing a new method for measuring its convergence; this allows for more widespread usage of a method that is inherently more valid at low frequencies than many other common algorithms. Finally, I explain a new method for the stabilization and practical implementation of a faster version of the gpspi method, which makes its overall lower-frequency validity compared to other methods more practical and economical.
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