Geophysical Applications of Hermite Distributed Approximating Functionals

The thesis is centered on the Hermite Distributed Approximating Functional (HDAF). The HDAF provides a controllable two-parameter delta sequence converging to the Dirac delta function with arbitrary accuracy. Under appropriate choice of parameters, the HDAF can be used 1) to approximate to a band-limited function, 2) to solve PDEs, 3) to remove undesired signal components as a band-pass filter. Four different kinds of geophysical problems are studied using HDAF-based methods. Firstly, the HDAF is used to augment the low-frequency component of seismic data. The lack of low frequencies, due to the limitations of the acquisition devices and sources, is one of the most significant problems for the seismic inversion problem. I propose and demonstrate a method, which combines the least square fitting using HDAF with a Gap Reduction procedure, to augment seismic data with large low-frequency data gap. Next, I take advantage of the HDAF to solve the fractional Laplacian viscoacoustic wave equation for seismic modeling and imaging. The attenuation effect of viscosity often significantly deteriorates the quality of seismic images. A constant Q fractional Laplacian wave equation is introduced to incorporate the attenuation effect of seismic waves both for seismic modeling and seismic imaging. However, this wave equation involves a fractional Laplacian with spatially varying exponent. This leads to complex wave number and extremely difficult to solve such a Partial Differential Equation at a reasonable computational cost. By combing the HDAF with a phase shift plus interpolation (PSPI) technique, I demonstrate that the viscoacoustic wave equation can be solved at a significantly reduced computational cost while still attaining high accuracy. A brief numerical study of attenuation is included. The last topic I focus on is to upscale sonic frequency well log data to seismic frequency data. In clastic sediments, knowledge of the heterogeneity can be used to detect the productive layers. I use the HDAF for an upscaling method for calculating the amplitude of the fluctuations C33, obtained from well log data. Compared to the conventional simple moving average upscaling method, the HDAF provides better results at the lower frequencies.