Compressive sensing in seismic exploration : an outlook on a new paradigm

Many seismic exploration techniques rely on the collection of massive data volumes that are subsequently mined for information during processing. While this approach has been extremely successful in the past, current efforts toward higher resolution images in increasingly complicated regions of the Earth continue to reveal fundamental shortcomings in our workflows. Chiefly amongst these is the so-called “curse of dimensionality” exemplified by Nyquist’s sampling criterion, which disproportionately strains current acquisition and processing systems as the size and desired resolution of our survey areas continues to increase. We offer an alternative sampling method leveraging recent insights from compressive sensing towards seismic acquisition and processing for data that, from a traditional point of view, are considered to be undersampled. The main outcome of this approach is a new technology where acquisition and processing related costs are decoupled from the stringent Nyquist sampling criterion. At the heart of our approach lies randomized incoherent sampling that breaks subsampling-related interferences by turning them into harmless noise, which we subsequently remove by promoting sparsity in a transform-domain. Acquisition schemes designed to fit into this regime no longer grow significantly in cost with increasing resolution and dimensionality of the survey area, but instead its cost ideally only depends on transform-domain sparsity of the expected data. Our contribution is twofold. First, we demonstrate by means of carefully designed numerical experiments that ideas from compressive sensing can be adapted to seismic acquisition. Second, we leverage the property that seismic data volumes are well approximated by a small percentage of curvelet coefficients. Thus curvelet-domain sparsity allows us to recover conventionally-sampled seismic data volumes from compressively-sampled data volumes whose size exceeds this percentage by only a small factor. Because compressive sensing combines transformation and encoding by a single linear encoding step, this technology is directly applicable to seismic acquisition and therefore constitutes a new paradigm where acquisitions costs scale with transform-domain sparsity instead of the gridsize. We illustrate this principle by showcasing recovery of a real seismic line from simulated compressively sampled acquisitions. [1] Part I of this paper is adapted from Randomized sampling and sparsity: getting more information from fewer samples. Geophysics 75, WB173 (2010); doi:10.1190/1.350614. [2] Seismic Laboratory for Imaging and Modeling, Department of Earth and Ocean Sciences, the University of British Columbia, 6339 Stores Road, Vancouver, V6T 1Z4, BC, Canada. Email: fherrmann@eos.ubc. ca 1 Outline In Part I, we showed by means of carefully designed experiments that the theory of compressive sensing can be successfully adapted to seismic-data acquisition. The experiments confirmed that accurate recoveries are possible from compressively sampled data volumes that exceed the size of conventionally compressed data volumes by only a small factor. In this part II, we adapt these findings to the recovery of a real seismic line from compressively-sampled ‘Land’ and ‘Marine’ data. Because compressive sensing combines transformation and encoding by a single linear encoding step, this technology is directly applicable to seismic acquisition and therefore constitutes a new paradigm where acquisitions costs scale with transform-domain sparsity instead of the grid size. We illustrate this principle by showcasing recovery of a real seismic line from simulated compressively sampled acquisitions. 2 An academic case study Now that we established that high SNR’s are achievable with modest oversampling ratios, we study the performance of our recovery algorithm on a seismic line from the Gulf of Suez by comparing two simultaneous-source scenarios with coincident source-receiver positions: • Land acquisition with random amplitude encoding: Here, sequential impulsive sources are replaced by impulsive simultaneous `phase-encoded’ sources. Mathematically, simultaneous measurements are obtained by replacing the sampling matrix for the sources—normally given by identity matrix—by a measurement matrix obtained by phase encoding along the source coordinate. Following (Romberg, 2009) and (Herrmann et al., 2009b), we define the measurement matrix by the following Kronecker product