Imaging and inversion with seismic data recorded with sources and receivers at the surface are powerful tools to infer knowledge about the subsurface. However, creating an image with seismic data is unfortunately not as easy as taking a picture with a smartphone. The estimated subsurface models in many situations are far from ideal due to the low quality nature of the data. One of the reasons can be weathering of the near-surface geology that generates unconsolidated material characterized by slow velocity with rapidly varying, heterogeneous and season-dependent nature. Acquiring seismic data on such near-surface leads to complex wave propagation, posing challenges to imaging and inversion. In this dissertation, we tackle the weathering effects during seismic data processing, imaging and inversion with low-rank-based methods.

One approach to tackle the weathering effects on seismic data is removing them during seismic data processing. To do so for 2D data, we propose a model-independent low-rank-based near-surface estimation and correction in the midpoint-offset-frequency domain. In this domain, ideal data exhibit low rank structures, which get destroyed due to the influence of the weathering layers. Accordingly, the method makes use of the redundant nature of seismic data that allows for accurate approximation by low-rank matrices. To estimate the time shifts that compensate for the weathering effects, we cross-correlate a data set influenced by the near-surface weathering layers with its low-rank approximated version. Since we estimate time shifts (commonly referred to as statics) and no longer the directly low-rank approximated data, we avoid losses of the amplitude information. To improve the estimated statics and to alleviate the need for accurate rank selection for low-rank approximation, we implement the method in an iterative and multi-scale fashion. Since the low-rank approximation deteriorates at high frequencies, we utilize its better performance at low frequencies and exploit the common statics amongst different frequency bands. Using synthetic and field data, we demonstrate the performance of the proposed proposed, which requires no knowledge of the subsurface model, demands minimal data pre-processing, and provides accurate solutions with high computational efficiency compared to existing techniques.

When seismic data acquired on complex near-surface are additionally subsampled for economical reasons, such as monitoring of sequestrated carbon dioxide and hydrogen, the problem is further exacerbated. Both the weathering layers and randomized subsampling render coherent energy incoherent. Therefore, they both contribute to destruction of the low-rank structure commonly associated with statics-free densely-sampled data. Frugal data acquisition in complex near-surface regimes makes separation of the distinct sampling and weathering effects on the rank structure difficult, which as a result lead to poor reconstruction. To overcome that, we propose to reconstruct the data with joint rank-reduction-based near-surface correction and interpolation. The method simultaneously accounts for the weathering and subsampling effects to provide accurate reconstruction. Since low-rank approximation is used for near-surface correction, we also utilize it in rank-minimization interpolation as a cost-free initial solution to the optimization problem. As both near-surface correction and interpolation operate in the midpoint-offset domain, we avoid the cost of transformations back and forth from the source-receiver to midpoint-offset transform domain. Consequently, the proposed reconstruction, which shows its potential on synthetic and field data, additionally increases the computational efficiency.

While the aforementioned near-surface correction deals with 2D data, the Earth is a 3D object that requires acquisition of 5D data for proper subsurface model estimation. For 5D data, the limitations and challenges of conventional near-surface correction methods are magnified. To avoid them, we propose a 5D model-independent low-rank-based near-surface correction. To compute the singular value decomposition of 5D data volumes with 1 temporal and 4 spatial dimensions, which is necessary for low-rank approximation, we need to perform matricization of the 5D data, i.e. organization of the 5D data into matrices. At the same time, it is essential that the chosen organization domain reveals the underlying low-rank structure. Therefore, we first analyze different matricization domains that can be used to organize the 5D data. Similar to the 2D case, we show that --- in the potential domain --- the near-surface weathering layers render coherent energy incoherent, which results in slowly decaying singular values compared to the statics-free data that are of low-rank nature. The proposed method, which we show on synthetic and field data, enjoys the same benefits of the proposed method for 2D data, in addition to being able to capture the 3D nature of the Earth.

Due to the complex nature of the near-surface and due to its impact on the subsurface model, the near-surface model gets treated separately from the subsurface model. However, the optimal goal is not to remove the near-surface effects with data processing, but to accurately estimate near- and sub-surface models simultaneously. To do so, we use the inherent scale separation of joint migration inversion that estimates a low-wavenumber velocity and high-wavenumber reflectivity. Since rapid variations in surface elevation and near-surface model result in high wavenumber effects, they end up affecting the reflectivity model. At the same time, the estimated reflectivity influences velocity estimation. Consequently, JMI provides erroneous subsurface models in the presence of complex weathering layers. To mitigate that, we use multi-scale low-rank updates in the reflectivity domain. The proposed method reduces the near-surface effects at the initial iterations, but it allows more details of the near-surface model to enter the solution at later iterations. In the end, we estimate accurate near- and sub-surface models simultaneously without the need to bypass the weathering layers.