A method for analyzing at least one characteristic of a geological formation may include obtaining measured data for the geological formation based upon a logging tool. Measured data may come from multiple passes or multiple depths of investigation. The method may further include generating a kernel describing a known linear mapping between the measured data and unknown data points representing at least one characteristic of the geological formation, and a redundant dictionary including a plurality of different basis functions expected to span the solution space of the unknown data points. The unknown data points representing the at least one characteristic of the geological formation may be determined from the measured data, the kernel and the redundant dictionary based upon an L1 minimization.