Mathematical modeling of seismic data: Multiscale noise removal and recognition algorithm

In this thesis, we present a novel artifact removal algorithm based on a seismic acquisition artifact data model, in which the artifact pattern is postulated to be additive to the actual geological signals. The algorithm can effectively remove artifacts that are highly correlated with the postulated pattern without affecting the spatial resolution. The main assumption of this algorithm is that the artifact is orthogonal to the underlying signal at certain scales of its DWT coefficients. The underlying signal is modeled as a sum of a short range dependent random noise and a deterministic signal. We prove that when the number of sampling points goes to infinity, the underlying signal and the artifact are asymptotically orthogonal to each other at a certain fine scale of its DWT coefficients. We name this algorithm as Wavelet Statistical Orthogonal Noise Reduction Scheme. We prove that with the WSO noise reduction scheme and for this noise model, when the number of sampling points goes to infinity, the average error between the evaluated artifact and the real artifact will go to zero and the probability of any non zero relative error between the estimated underlying signal and the real underlying signal will go to zero. Therefore, we know that the bias introduced from the orthogonality assumption will be reduced to zero as the number of sampling points goes to infinity. We use numerical experiments to verify that the above assumption is statistically true especially for large data sets, and we also use both synthetic and real seismic data sets to demonstrate the performance of this algorithm.