In ocean acoustics and seismology, the Earth’s subsurface is imaged using acoustic and seismic waves. As they propagate through the ocean and solid earth, these waves obtain geophysical information. This information is recovered via optimization procedures which fit physical models to wavefield observations from sensor arrays. The estimation of geophysical model parameters from the observations, typically referred to as inverse problems, are challenging due to many issues, e.g. noisy and incomplete observations, as well as non-linear forward models.
In this dissertation, geophysical inversion methods are developed based on sparse model- ing and dictionary learning, an unsupervised machine learning method. These techniques employ more sophisticated model priors and latent representations than conventional methods, and obtain state-of-the-art performance in a variety of signal processing tasks. Sparse modeling assumes that signals can be reconstructed to acceptable accuracy using a small (sparse) number of vectors, called atoms, from a larger set of atoms, or dictionary. Sparsifying dictioniaries can be designed from generic functions such as wavelets, or can be learned directly from the data via dictionary learning. Provided sufficient signal examples exist, dictionary learning can learn sparsifying dictionaries which obtain better performance than generic dictionaries. Conventional methods, rely on smoothness and second order statistics (e.g. empirical orthogonal functions (EOFs)) to estimate geophysical structure. In contrast, sparse methods potentially permit the recovery of true smooth and discontinuous geophysical structures.
Ocean acoustic sound speed profile (SSP) estimation requires the inversion of acoustic fields using limited observations. A specific case of sparse modeling, called compressive sensing (CS) asserts that certain underdetermined problems can be solved in high resolution, provided their solutions are sparse. CS is used to estimate SSPs in a range-independent shallow ocean by inverting a non-linear acoustic propagation model. It is shown that SSPs can be estimated using CS to resolve fine-scale structure.
To provide constraints on their inversion, ocean sound speed profiles (SSPs) are modeled often using empirical orthogonal functions (EOFs). However, this regularization, which uses the leading order EOFs with a minimum-energy constraint on their coefficients, often yields low resolution SSP estimates. It is shown that dictionary learning, a form of unsupervised machine learning, can improve SSP resolution by generating a dictionary of shape functions for sparse modeling that optimally compress SSPs; both minimizing the reconstruction error and the number of coefficients. These learned dictionaries (LDs) are not constrained to be orthogonal and thus, fit the given signals such that each signal example is approximated using few LD entries. LDs describing SSP observations from the High Frequency ‘97 experiment and the South China Sea are generated using the K-SVD algorithm. These LDs better explain SSP variability and require fewer coefficients than EOFs, describing much of the variability with one coefficient. Thus, LDs improve the resolution of SSP estimates with negligible computational burden.
A 2D travel time tomography method is developed based on sparse modeling and dictionary learning. The method regularizes the inversion by modeling groups of slowness pixels from discrete slowness maps, called patches, as sparse linear combinations of atoms from a dictionary. Dictionary learning is used in the inversion method to adapt dictionaries to specific slowness maps. This patch regularization, called the local model, is integrated into the overall slowness map, called the global model. The local model considers small-scale variations using a sparsity constraint and the global model considers larger-scale features constrained using l2 regularization. This strategy in a locally-sparse travel time tomography (LST) approach enables simultaneous modeling of smooth and discontinuous slowness features. This is in contrast to conventional tomography methods, which constrain models to be exclusively smooth or discontinuous. We develop a maximum a posteriori formulation for LST and exploit the sparsity of slowness patches using dictionary learning. The LST approach compares favorably with smoothness and total variation regularization methods on densely, but irregularly sampled synthetic slowness maps.
Finally, the LST travel time tomography method is used to obtain high-resolution subsur- face geophysical structure in Long Beach, CA, from seismic noise recorded on a “large-N” array with 5200 geophones (∼ 13.5 million travel times). LST exploits the dense sampling obtained by ambient noise processing on large arrays by learning a dictionary of local, or small-scale, geophysical features directly from the data. Using LST, a high-resolution 1 Hz Rayleigh wave phase speed map of Long Beach is obtained. Among the geophysical features shown in the map, the important Silverado aquifer is well isolated relative to previous surface wave tomography studies. The results show promise for LST in obtaining detailed geophysical structure in travel time tomography studies.