Imaging and inversion of land seismic data affected by complex weathering layers near the surface are challenging. When the data are additionally subsampled for economical reasons such as monitoring of sequestrated carbon dioxide and hydrogen, the problem is further exacerbated due to the combined influence of subsampling and weathering layers. First, interpolation performs poorly because the weathering layers reduce the data’s coherency. Second, near-surface corrections require knowledge of the subsurface model, separation between primaries and multiples, as well as subsurface velocity estimation, which are difficult to perform from subsampled data. To overcome these hurdles, we combine seismic interpolation and statics estimation into a joint single rank-reduction-based algorithm. To our knowledge, this is the first time that this has been done. Our method simultaneously accounts for the weathering and subsampling effects, which both contribute to the low-rank (LR) structure destruction typically associated with statics-free densely sampled data, to provide accurate reconstruction. Because an LR approximation is used for statics estimation, we also use it in rank-minimization interpolation as a cost-free initial solution to the optimization problem. As statics estimation and interpolation operate in the midpoint-offset domain, we avoid the cost of transformations back and forth from the source-receiver to the midpoint-offset transform domain. Consequently, our reconstruction, which indicates its potential on synthetic and field data, also is computationally efficient.@en

Surface consistency forms the basis for short-wavelength statics estimation. When raypaths in the near surface diverge from a normal incidence or when the normal moveout (NMO) velocity is inaccurate, surface-consistent methods may fail to estimate accurate statics. Existing nonsurface-consistent techniques can be prone to errors due to the need to construct pilot traces or pick horizons while imposing additional computational costs. To overcome these limitations and correct for the surface- and nonsurface-consistent statics, we have developed a low-rank-based residual statics (LR-ReS) estimation and correction framework. The method makes use of the redundant nature of seismic data by using its low-rank structure in the midpoint-offset-frequency domain. Due to the near-surface effect, the low-rank structure is destroyed. Therefore, we estimate the statics by means of low-rank approximation and crosscorrelation. To alleviate the need for accurate rank selection for low-rank approximation and improved statics estimation, we implement the method in an iterative and multiscale fashion. Because the low-rank approximation deteriorates at high frequencies, we use its better performance at low frequencies and exploit the common statics among the different frequency bands. The LR-ReS estimation and correction can be applied to data without an NMO correction, which makes statics estimation independent of the NMO velocity errors. Consequently, it can reduce the multiple iterations of the NMO velocity estimation and short-wavelength statics correction commonly needed for conventional methods to improve their performance. Moreover, the LR-ReS estimation does not require windowing of a noise-free area containing aligned primaries or mute to avoid the NMO stretch effect, which enables statics correction of the wavefield of all offsets. To evaluate the performance of our method, we apply it to simulated data and a challenging field data set affected by complex weathering layers and noise, which indicate a substantial improvement compared with conventional short-wavelength statics correction.

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Surface-consistent residual statics correction for land seismic data does not account for the source - receiver offset. Consequently, it requires normal moveout (NMO) corrected gathers to bring raypaths close to the normal incidence. When the NMO velocity is inaccurate or unavailable, the estimated statics suffer. Therefore, multiple passes of NMO velocity picking and residual statics estimation become essential, which are efforts and time consuming. To avoid this, we utilize a rank-based solution that is capable of estimating non-surface-consistent residual statics. The method is based on the rank property of frequency slices in the midpoint-offset domain, where ideal seismic data is of low-rank nature, while data with residual statics exhibits higher rank. Accordingly, we estimate the statics that lead to the desired low-rank signal via means of low-rank approximation and cross-correlation in an iterative and multi-rank-scale approach. Since we estimate non-surface-consistent statics by accounting for the offset of each trace, it is no longer required to have NMO corrected gathers. Consequently, the method does not require windowing over a noise-free section containing primaries or windowing to avoid the NMO stretch effect, which are required by conventional residual statics correction. Numerical results on simulated and field data suggest that the method has the potential of replacing existing residual statics correction techniques.

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Most short-wavelength statics-correction methods are based on the surface-consistency assumption depending on locations of sources and receivers at the surface. Even though this assumption may work in practice, it is not the most accurate solution since raypaths in the near-surface are not strictly vertical. Existing non-surface-consistent residual statics correction methods require constructing pilot traces or picking horizons, which make them prone to errors. We propose a low rank-based residual statics correction framework (LR-Res) that estimates non-surface-consistent statics from monochromatic frequency slices in the midpointoffset transform domain in an iterative and multi-scale fashion. When land seismic data contains gaps, an additional layer of complexity arises because interpolating the data suffers as residual statics break the data's continuity. To reconstruct densely-sampled data, we utilize ideas from our proposed LR-Res framework to jointly correct for short-wavelength statics and interpolate the data. We demonstrate the performance of our proposed methods in accurately correcting for non-surface-consistent, short-wavelength statics of densely sampled data, as well as reconstructing densely sampled data from subsampled data affected by short-wavelength statics, which we also compare with conventional methods.

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