The development of new tools for high-resolution seismic imaging has been for many years one of the key challenges faced by earthquake and exploration seismologists. In order to make data amenable to imaging analysis, pre-processing steps are of great importance. This thesis proposes a new method for pre-processing teleseismic data based on the linear radon transform implemented according to compressive sensing theory – a novel theory about acquiring and recovering the sparsest signals (with minimum significant coefficients) in the most efficient way possible with the help of incoherent measurements. The LRT works by mapping data into a sparsity-promoting domain (called the radon domain) where the desired signals can be easily isolated, classified, filtered and enhanced; and where noise can be attenuated or completely removed. The performance of the LRT is enhanced in terms of both high-resolution and computational cost by formulating the problem as an inverse problem in the frequency domain.

This work shows that, unlike the common wisdom, irregularity in spatial sampling of teleseismic wavefields can be beneficial because it provides the incoherency needed to solve the compressive sensing problem and therefore recover the sparsest solutions in the radon domain. The inverse problem formulation yields the added advantage of automatic spatial interpolation and phase isolation after data reconstruction, and enables to regularize the problem by imposing sparsity constraint (instead of smoothness, which is the constraint usually adopted). We discuss and investigate the resolving power and applicability of convex and non-convex types of regularizers inspired from compressive sensing theory, and we establish a lower bound on the number of measurements needed to resolve certain time dips related to signals of interest within the data. We finish by applying the method to synthetic and recorded datasets and show how we do signal extraction, noise removal and spatial interpolation on teleseismic wavefields.