Compressive sensing, sparse transforms and deblending
Daniel O. Trad
Data acquisition is by far the most expensive and problematic part of seismic methods. In particular, 3-dimensional data surveys always have deficient sampling in at least 2 of the 4 spatial dimensions. As a consequence, geophysicists make extensive efforts in the mitigation of sampling problems. These efforts usually involve two directions: data interpolation and simultaneous acquisition. Interpolation is intended to create new seismic traces from the acquired samples by using sparse transformations. Simultaneous acquisition, also known as blending, attempts to mitigate the sampling problem by acquiring more data without increasing the acquisition cost. Simultaneous acquisition is a very costeffective approach that reduces the cost of seismic information in both marine and land settings. Its main difficulty is the processing of the resulting seismic data, which requires shot separation or deblending, very early in the signal processing chain.
In the last few years, the two approaches have been merged in geophysics with the name of Compressive Sensing (CS). CS refers to an approach developed in the field of mathematics, which permits to obtain information with less sampling by relying on the combination of irregular sampling and sparseness to extract information from sparsely sampled data. This name is a bit unfortunate, because in reality CS, as used in seismic, is a combination of sparse transforms plus deblending, and both technologies have been performed in geophysics much before Compressive Sensing existed. Nonetheless, the name has now stuck in geophysics, and involves acquiring data in a random fashion, using simultaneous sources, and performing deblending and denoising right at the beginning of processing by using sparse transforms.
In this paper, I will discuss the relationships between CS and sparse transforms, showing that both are just the same approach with a different name. Then, I will discuss one particular approach for deblending based on migration/demigration as the transform operator. Finally, I will consider the merge of 5D interpolation with LSMIG as a single approach for deblending.