Seismic surveys involve an artificial source of waves and a grid of receivers at the surface. Often, receivers could be missing either because they malfunctioned or could not be placed in certain locations. It could also be the fact that a local source of noise renders a receiver’s output as unusable. These gaps in the data cause problems in later stages of the seismic signal processing work flow via aliasing or incoherent noise and thus signal reconstruction is necessary. Modern algorithms utilise the principle of Compressive Sensing (CS) for reconstruction which uses the assumption that the signal of interest is either sparse in nature or in some other bases. Most algorithms are designed with the only aim to fill in gaps in the data without any consideration of learning bases or quantifying uncertainty in their predictions.

In this thesis, we approach the seismic CS problem using probabilistic data-driven models that are adaptable to seismic data. We propose to use algorithms from the Bayesian statistics and machine learning field that allow the construction of models using probability distributions over random variables. This allows the modelling of sparsity and provides flexibility by adding or removing basis functions from the model. It also provides the framework for learning new dictionaries of bases, associating uncertainty for each prediction and denoising seismic signals. More specifically, we utilise two Bayesian algorithms for seismic CS, the Relevance Vector Machine (RVM) and the Beta Process Factor Analysis (BPFA).

The RVM uses a sparsity promoting distribution over the coefficients of a linear combination of basis functions. By learning the appropriate parameters, the algorithm infers a predictive mean and predictive variance that is used for prediction of receivers’ values and uncertainty quantification. Experiments and comparisons on various seismic data show the effectiveness of the RVM with state-of-the-art reconstruction accuracy. Furthermore, its predictive variance is used along with modifications in order to create uncertainty maps with varying levels of correlation between uncertainty and respective reconstruction error of receivers.

On the other hand, BPFA uses an alternative approach to enforce sparsity providing exact zero coefficients as opposed to the RVM. Another advantage is that it also learns the bases from the available data and provides denoising of seismic signals. Experiments and comparisons on seismic data show that the BPFA obtains state-of-the-art reconstruction accuracy on various domains. In addition, the learned bases are used by other algorithms to improve their performance. An analysis of the BPFA’s inference procedure is given along with insights to reduce its computational cost. We also utilise the probabilistic nature of the BPFA and calculate the variance of the receivers’ predictions obtained during inference. Using this, we create uncertainty maps that are highly correlated with the reconstruction error, obtaining better results than the RVM’s predictive variance. Finally, an analysis of seismic signals with different levels of variance is undertaken in order to provide guidance for the best choice of algorithm per region.

The amount of seismic data available is growing, nevertheless quantity does not directly translate to quality. This creates the challenge to analyse and extract as much information and insight as possible. Using probabilistic data-driven models, we show how to achieve this by reconstructing seismic signals from under-sampled data, learn features from training data, denoise and create uncertainty maps for predictions in seismic surveys.