Processing seismic data in the presence of residual statics

Many common processing steps are degraded by static shifts in the data. The effects of static shifts are analogous to noise or missing samples in the data, and therefore can be treated using constraints on sparsity or simplicity. In this paper we show that random static shifts decrease sparsity in the Fourier and Radon transforms, as well as increase the rank of seismic data. We also show that the concepts of sparsity promotion and rank re-duction can be used to solve for static shifts as well as to carry out conventional processes in the presence of statics. The first algorithm presented is a modification to the reinsertion step of Projection Onto Convex Sets (POCS) and Tensor Completion (TCOM) that allows for the compensation of residual statics during 5D denoising and interpolation of seismic data. The method allows preserving residual statics during denoising, or correction of residual statics in the case of simultaneous de-noising and interpolation. An example is shown for a 5D re-construction of synthetic data with added noise, missing traces and random static shifts, as well as for a 2D stacked section with missing traces and static shifts. While standard recon-struction struggles in the presence of even small static shifts, reconstruction with simultaneous estimation of statics is able to accurately reconstruct the data. The second algorithm pre-sented is a Statics Preserving Sparse Radon transform (SPSR). This algorithm includes statics in the Radon bases functions, allowing for a sparse representation of statics-contaminated data in the Radon domain