A. M. Alfaraj
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1
Tackling the weathering with low ranks
Handling the complex near surface of land seismic data 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. ...
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.
Towards 3D near-surface correction without NMO
A rank-based approach
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.
In the presence of near-surface weathering layers, wave propagation may become complex and accurate velocity estimation can be challenging. As a result, reverse-time migration (RTM) and least-squares (LS)-RTM may provide inaccurate images of low-resolution contaminated with artifacts. Therefore, prior data conditioning with near-surface correction is routinely applied on land data. However, there is an associated risk that this process may also remove details from the subsurface model or may not fully account for the near-surface effects leading to poor and erroneous image. We propose to utilize a data-driven rank-based approach that does not require a velocity model to mitigate the effects of rapid changes of surface-elevation and properties of the weathering layers prior to RTM and LS-RTM. We demonstrate that this pre-processing step can restore the high-resolution information and reduce the artifacts of the migrated image while preserving the subsurface structure.
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.
An important imaging challenge is creating reliable seismic images without internal multiple crosstalk, especially in cases with strong overburden reflectivity. Several data-driven methods have been proposed to attenuate the internal multiple crosstalk, for which fully sampled data in the source and receiver side are usually required. To overcome this acquisition constraint, model-driven full-wavefield migration (FWM) can automatically include internal multiples and only needs dense sampling in either the source or receiver side. In addition, FWM can correct for transmission effects at the reflecting interfaces. Although FWM has been shown to work effectively in compensating for transmission effects and suppressing internal multiple crosstalk compared to conventional least-squares primary wavefield migration (PWM), it tends to generate relatively weaker internal multiples during modeling. Therefore, some leaked internal multiple crosstalk can still be observed in the FWM image, which tends to blend in the background and can be misinterpreted as real geology. Thus, we adopted a novel framework using local primary-and-multiple orthogonalization (LPMO) on the FWM image as a postprocessing step for leaked internal multiple crosstalk estimation and attenuation. Due to their opposite correlation with the FWM image, a positive-only LPMO weight can be used to estimate the leaked internal multiple crosstalk, whereas a negative-only LPMO weight indicates the transmission effects that need to be retained. Application to North Sea field data validates the performance of the proposed framework for removing the weak but misleading leaked internal multiple crosstalk in the FWM image. Therefore, with this new framework, FWM can provide a reliable solution to the long-standing issue of imaging primaries and internal multiples automatically, with proper primary restoration.
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.
Short-wavelength statics resulting from the unconsolidated near-surface weathering layers need to be corrected for in order to obtain an undistorted image of the subsurface. Existing methods based on the surface consistency assumption can be computationally intensive and may not fully correct for residual statics. In this work, we follow a data-driven and computationally efficient low-rank approximation approach to correct for short-wavelength statics. The method is based on the property that short-wavelength statics increase the rank of the data, while statics-free data is of low-rank nature in a transform domain. We demonstrate the performance of the proposed method on real land seismic data and compare it with conventional residual statics correction.