Seismic data (traces) usually demonstrate high correlation. We propose a scheme based on online dictionary learning, which explores the resemblance among local seismic traces to facilitate compression for communication. In order to alleviate the transmission overhead caused by the slow convergence of online dictionary scheme, sparse constraints and a sliding window mechanism are applied to the incremental components of the dictionaries, which significantly improve the performance of online dictionary learning scheme in the sense of communication cost.

Full waveform inversion (FWI) delivers high-resolution images of a subsurface medium property by minimizing itera-tively the misfit between observed and simulated seismic data, and is commonly used by the oil and gas industry for geophysical exploration. FWI is a challenging problem because seismic surveys cover ever larger areas of interest and collect massive volumes of data. The dimensionality of the problem and the heterogeneity of the medium both stress the need for faster algorithms, so sparse regularization techniques can be used to accelerate and improve imaging results. In this paper, we propose a compressive sensing method for the FWI problem by exploiting the sparsity of geological model perturbations over learned dictionaries. Based on stochastic approximations, the dictionaries are updated itera-tively to adapt changing models during FWI iterations. Meanwhile, the dictionaries are kept orthonormal in order to maintain the corresponding transform in a fast and compact manner so that these transforms do not introduce extra computational overhead to FWI. Establishing such a sparsity regu-larization on the model enables us to significantly reduce the workload by only collecting 0.625% of the field data without introducing subsampling artifacts. Hence, the computational burden of large-scale FWI problems can be greatly reduced.

In this paper, we examine adaptive compression policies, when the sequence of vector-valued measurements to be compressed is noisy and the compressed variables are themselves noisy. The optimization criterion is information gain. In the case of sequential scalar compressions, the unit-norm compression vectors that greedily maximize per-stage information gain are eigenvectors of an a priori error covariance matrix, and the greedy policy selects them according to eigenvalues of a posterior covariance matrix. These eigenvalues depend on all previous compressions and are computed recursively. A water-filling solution is given for the optimum compression policy that maximizes net information gain, under a constraint on the average norm of compression vectors. We provide sufficient conditions under which the greedy policy for maximizing stepwise information gain actually is optimal in the sense of maximizing the net information gain. In the case of scalar compressions, our examples and simulation results illustrate that the greedy policy can be quite close to optimal when the noise sequences are white.

The general theory of greedy approximation is well developed. Much less is known about how specific features of a dictionary can be used to our advantage. In this paper, we discuss incoherent dictionaries. We build a new greedy algorithm which is called the orthogonal super greedy algorithm (OSGA). We show that the rates of convergence of OSGA and the orthogonal matching pursuit (OMP) with respect to incoherent dictionaries are the same. Based on the analysis of the number of orthogonal projections and the number of iterations, we observed that OSGA is times simpler (more efficient) than OMP. Greedy approximation is also a fundamental tool for sparse signal recovery. The performance of orthogonal multimatching pursuit, a counterpart of OSGA in the compressed sensing setting, is also analyzed under restricted isometry property conditions.