TCCS -- TABLE OF CONTENTS

TCCS -- TABLE OF CONTENTS

Selecting an optimal aperture in Kirchhoff migration using dip-angle images [pdf 1.5M]
Alexander Klokov and Sergey Fomel
We present a method for selecting a migration aperture in Kirchhoff migration. We first split migrated data into constant-dip-angle partial images. Then, in every partial image, we estimate the consistency between each event and the constant dip of the analyzed section. We filter out events whose slope is far from the corresponding dip. Stacking of the filtered partial images corresponds to migration having an optimal aperture. Synthetic and real data examples demonstrate that the proposed approach to migration-aperture optimization is able to reduce migration noise while preserving diffraction energy, which characterizes small geological objects and brings additional resolution to the image.
A reversible transform for seismic data processing [pdf 696K]
William Burnett and Robert Ferguson
We use the nonstationary equivalent of the Fourier shift theorem to derive a general one-dimensional integral transform for the application and removal of certain seismic data processing steps. This transform comes from the observation that many seismic data processing steps can be viewed as nonstationary shifts. The continuous form of the transform is exactly reversible, and the discrete form provides a general framework for unitary and pseudounitary imaging operators. Any processing step which can be viewed as a nonstationary shift in any domain is a special case of this transform. Nonstationary shifts generally produce coordinate distortions between input and output domains, and those that preserve amplitudes do not conserve the energy of the input signal. The nonstationary frequency and time distortions and nonphysical energy changes inherent to such operations are predicted and quantified by this transform. Processing steps of this type are conventionally implemented using interpolation operators to map discrete data values between input and output coordinate frames. Although not explicitly derived to perform interpolation, the transform here assumes the Fourier basis to predict values of the input signal between sampling locations. We demonstrate how interpolants commonly used in seismic data processing and imaging approximate the proposed method. We find that our transform is equivalent to the conventional sinc-interpolant with no truncation. Once the transform is developed, we demonstrate its numerical implementation by matrix-vector multiplication. As an example, we use our transform to apply and remove normal moveout.
Seismic wave extrapolation using lowrank symbol approximation [pdf 2.1M]
Sergey Fomel, Lexing Ying, and Xiaolei Song
We consider the problem of constructing a wave extrapolation operator in a variable and possibly anisotropic medium. Our construction involves Fourier transforms in space combined with the help of a lowrank approximation of the space-wavenumber wave-propagator matrix. A lowrank approximation implies selecting a small set of representative spatial locations and a small set of representative wavenumbers. We present a mathematical derivation of this method, a description of the lowrank approximation algorithm, and numerical examples which confirm the validity of the proposed approach. Wave extrapolation using lowrank approximation can be applied to seismic imaging by reverse-time migration in 3D heterogeneous isotropic or anisotropic media.
OC-seislet: seislet transform construction with differential offset continuation [pdf 3.2M]
Yang Liu and Sergey Fomel
Many of the geophysical data analysis problems, such as signal-noise separation and data regularization, are conveniently formulated in a transform domain, where the signal appears sparse. Classic transforms such as the Fourier transform or the digital wavelet transform, fail occasionally in processing complex seismic wavefields, because of the nonstationarity of seismic data in both time and space dimensions. We present a sparse multiscale transform domain specifically tailored to seismic reflection data. The new wavelet-like transform - the OC-seislet transform - uses a differential offset-continuation (OC) operator that predicts prestack reflection data in offset, midpoint, and time coordinates. It provides high compression of reflection events. In the transform domain, reflection events get concentrated at small scales. Its compression properties indicate the potential of OC-seislets for applications such as seismic data regularization or noise attenuation. Results of applying the method to both synthetic and field data examples demonstrate that the OC-seislet transform can reconstruct missing seismic data and eliminate random noise even in structurally complex areas.
Stratigraphic coordinates, a coordinate system tailored to seismic interpretation [pdf 1.5M]
Parvaneh Karimi and Sergey Fomel
In certain seismic data processing and interpretation tasks, such as spiking deconvolution, tuning analysis, impedance inversion, spectral decomposition, etc., it is commonly assumed that the vertical direction is normal to reflectors. This assumption is false in the case of dipping layers and may therefore lead to inaccurate results. To overcome this limitation, we propose a coordinate system in which geometry follows the shape of each reflector and the vertical direction corresponds to normal reflectivity. We call this coordinate system stratigraphic coordinates. We develop a constructive algorithm that transfers seismic images into the stratigraphic coordinate system. The algorithm consists of two steps. First, local slopes of seismic events are estimated by plane-wave destruction; then structural information is spread along the estimated local slopes, and horizons are picked everywhere in the seismic volume by the predictive-painting algorithm. These picked horizons represent level sets of the first axis of the stratigraphic coordinate system. Next, an upwind finite-difference scheme is used to find the two other axes, which are perpendicular to the first axis, by solving the appropriate gradient equations. After seismic data are transformed into stratigraphic coordinates, seismic horizons should appear flat, and seismic traces should represent the direction normal to the reflectors. Immediate applications of the stratigraphic coordinate system are in seismic image flattening and spectral decomposition. Synthetic and real data examples demonstrate the effectiveness of stratigraphic coordinates.
Lowrank finite-differences and lowrank Fourier finite-differences for seismic wave extrapolation in the acoustic approximation [pdf 1.2M]
Xiaolei Song, Sergey Fomel, and Lexing Ying
We introduce a novel finite-difference (FD) approach for seismic wave extrapolation in time. We derive the coefficients of the finite-difference operator from a lowrank approximation of the space-wavenumber, wave-propagator matrix. Applying the technique of lowrank finite-differences, we also improve the finite difference scheme of the two-way Fourier finite differences (FFD). We call the new operator lowrank Fourier finite differences (LFFD). Both the lowrank FD and lowrank FFD methods can be applied to enhance accuracy in seismic imaging by reverse-time migration. Numerical examples confirm the validity of the proposed technique.
Accelerated plane-wave destruction [pdf 1.2M]
Zhonghuan Chen, Sergey Fomel, and Wenkai Lu
When plane-wave destruction (PWD) is implemented by implicit finite differences, the local slope is estimated by an iterative algorithm. We propose an analytical estimator of the local slope that is based on convergence analysis of the iterative algorithm. Using the analytical estimator, we design a noniterative method to estimate slopes by a three-point PWD filter. Compared with the iterative estimation, the proposed method needs only one regularization step, which reduces computation time significantly. With directional decoupling of the plane-wave filter, the proposed algorithm is also applicable to 3D slope estimation. We present both synthetic and field experiments to demonstrate that the proposed algorithm can yield a correct estimation result with shorter computational time.
On anelliptic approximations for $qP$ velocities in transversally isotropic media [pdf 504K]
Sergey Fomel
I develop a unified approach for approximating phase and group velocities of $qP$ seismic waves in a transversally isotropic medium with the vertical axis of symmetry (VTI). While the exact phase velocity expressions involve four independent parameters to characterize the elastic medium, the proposed approximate expressions use only three parameters. This makes them more convenient for use in surface seismic experiments, where estimation of all the four parameters is problematic. The three-parameter phase-velocity approximation coincides with the previously published ``acoustic'' approximation of Alkhalifah. The group velocity approximation is `new and noticeably more accurate than some of the previously published approximations. I demonstrate an application of the group velocity approximation for finite-difference computation of traveltimes.
Iterative deblending of simultaneous-source seismic data using seislet-domain shaping regularization [pdf 4.9M]
Yangkang Chen, Sergey Fomel, and Jingwei Hu
We introduce a novel iterative estimation scheme for separation of blended seismic data from simultaneous sources. The scheme is based on an augmented estimation problem, which can be solved by iteratively constraining the deblended data using shaping regularization in the seislet domain. We formulate the forward modeling operator in the common receiver domain, where two sources are assumed to be blended using a random time-shift dithering approach. The nonlinear shaping-regularization framework offers some freedom in designing a shaping operator to constrain the model in an underdetermined inverse problem. We design the backward operator and the shaping operator for the shaping regularization framework. The backward operator can be optimally chosen as a half of the identity operator in the two-source case, and the shaping operator can be chosen as coherency-promoting operator. Three numerically blended synthetic datasets and one numerically blended field dataset demonstrate the high-performance deblending effect of the proposed iterative framework. Compared with alternative $f-k$ domain thresholding and $f-x$ predictive filtering, seislet-domain soft thresholding exhibits the most robust behavior.
Seismic data decomposition into spectral components using regularized nonstationary autoregression [pdf 1.2M]
Sergey Fomel
Seismic data can be decomposed into nonstationary spectral components with smoothly variable frequencies and smoothly variable amplitudes. To estimate local frequencies, I use a nonstationary version of Prony's spectral analysis method defined with the help of regularized nonstationary autoregression (RNAR). To estimate local amplitudes of different components, I fit their sum to the data using regularized nonstationary regression (RNR). Shaping regularization ensures stability of the estimation process and provides controls on smoothness of the estimated parameters. Potential applications of the proposed technique include noise attenuation, seismic data compression, and seismic data regularization.
Theory of 3-D angle gathers in wave-equation seismic imaging [pdf 380K]
Sergey Fomel
I present two methods for constructing angle gathers in 3-D seismic imaging by downward extrapolation. Angles in angle gathers refer to the scattering angle at the reflector and provide a natural access to analyzing migration velocity and amplitudes. In the first method, angle gathers are extracted at each downward-continuation step by mapping transformations in constant-depth frequency slices. In the second method, one extracts angle gathers after applying the imaging condition by transforming local offset gathers in the depth domain. The second approach generalizes previously published algorithms for angle-gather construction in 2-D and common-azimuth imaging.
A fast butterfly algorithm for generalized Radon transforms [pdf 2.1M]
Jingwei Hu, Sergey Fomel, Laurent Demanet, and Lexing Ying
Generalized Radon transforms such as the hyperbolic Radon transform cannot be implemented as efficiently in the frequency domain as convolutions, thus limiting their use in seismic data processing. We introduce a fast butterfly algorithm for the hyperbolic Radon transform. The basic idea is to reformulate the transform as an oscillatory integral operator and to construct a blockwise low-rank approximation of the kernel function. The overall structure follows the Fourier integral operator (FIO) butterfly algorithm. For two-dimensional data, the algorithm runs in complexity $O(N^2N)$ , where $N$ depends on the maximum frequency and offset in the dataset and the range of parameters (intercept time and slowness) in the model space. Using a series of examples, we show that the proposed algorithm can be significantly more efficient than the conventional time-domain integration.
Seismic data interpolation beyond aliasing using regularized nonstationary autoregression [pdf 1.9M]
Yang Liu and Sergey Fomel
Seismic data are often inadequately or irregularly sampled along spatial axes. Irregular sampling can produce artifacts in seismic imaging results. We present a new approach to interpolate aliased seismic data based on adaptive prediction-error filtering (PEF) and regularized nonstationary autoregression. Instead of cutting data into overlapping windows (patching), a popular method for handling nonstationarity, we obtain smoothly nonstationary PEF coefficients by solving a global regularized least-squares problem. We employ shaping regularization to control the smoothness of adaptive PEFs. Finding the interpolated traces can be treated as another linear least-squares problem, which solves for data values rather than filter coefficients. Compared with existing methods, the advantages of the proposed method include an intuitive selection of regularization parameters and fast iteration convergence. Benchmark synthetic and field data examples show that the proposed technique can successfully reconstruct data with decimated or missing traces.
Local seismic attributes [pdf 140K]
Sergey Fomel
Local seismic attributes measure seismic signal characteristics not instantaneously at each signal point and not globally across a data window but locally in the neighborhood of each point. I define local attributes with the help of regularized inversion and demonstrate their usefulness for measuring local frequencies of seismic signals and local similarity between different datasets. I use shaping regularization for controlling the locality and smoothness of local attributes. A multicomponent image registration example from a nine-component land survey illustrates practical applications of local attributes for measuring differences between registered images.
Shaping regularization in geophysical estimation problems [pdf 1.4M]
Sergey Fomel
Regularization is a required component of geophysical estimation problems that operate with insufficient data. The goal of regularization is to impose additional constraints on the estimated model. I introduce shaping regularization, a general method for imposing constraints by explicit mapping of the estimated model to the space of admissible models. Shaping regularization is integrated in a conjugate-gradient algorithm for iterative least-squares estimation. It provides the advantage of a better control on the estimated model in comparison with traditional regularization methods and, in some cases, leads to a faster iterative convergence. Simple data interpolation and seismic velocity estimation examples illustrate the concept.
Kirchhoff migration using eikonal-based computation of traveltime source-derivatives [pdf 1.4M]
Siwei Li and Sergey Fomel
The computational efficiency of Kirchhoff-type migration can be enhanced by employing accurate traveltime interpolation algorithms. We address the problem of interpolating between a sparse source sampling by using the derivative of traveltime with respect to the source location. We adopt a first-order partial differential equation that originates from differentiating the eikonal equation to compute the traveltime source-derivatives efficiently and conveniently. Unlike methods that rely on finite-difference estimations, the accuracy of the eikonal-based derivative does not depend on input source sampling. For smooth velocity models, the first-order traveltime source-derivatives enable a cubic Hermite traveltime interpolation that takes into consideration the curvatures of local wave-fronts and can be straight-forwardly incorporated into Kirchhoff anti-aliasing schemes. We provide an implementation of the proposed method to first-arrival traveltimes by modifying the fast-marching eikonal solver. Several simple synthetic models and a semi-recursive Kirchhoff migration of the Marmousi model demonstrate the applicability of the proposed method.
Seismic data analysis using local time-frequency decomposition [pdf 1.8M]
Yang Liu and Sergey Fomel
Many natural phenomena, including geologic events and geophysical data, are fundamentally nonstationary - exhibiting statistical variation that changes in space and time. Time-frequency characterization is useful for analyzing such data, seismic traces in particular. We present a novel time-frequency decomposition, which aims at depicting the nonstationary character of seismic data. The proposed decomposition uses a Fourier basis to match the target signal using regularized least-squares inversion. The decomposition is invertible, which makes it suitable for analyzing nonstationary data. The proposed method can provide more flexible time-frequency representation than the classical S transform. Results of applying the method to both synthetic and field data examples demonstrate that the local time-frequency decomposition can characterize nonstationary variation of seismic data and be used in practical applications, such as seismic ground-roll noise attenuation and multicomponent data registration.
Time-frequency analysis of seismic data using synchrosqueezing wavelet transform [pdf 2.4M]
Yangkang Chen, Tingting Liu, Xiaohong Chen, Jingye Li, and Erying Wang
Time-frequency (TF) decomposition is used for characterizing the non-stationary relation between time and instantaneous frequency, which is very important in the processing and interpretation of seismic data. The conventional time-frequency analysis approaches suffer from the contradiction between time resolution and frequency resolution. A new time-frequency analysis approach is proposed based on the synchrosqueezing wavelet transform (SSWT). The SSWT is an empirical-mode-decomposition-like tool but uses a different approach in constructing the components. With the help of the synchrosqueezing techniques, the SSWT can obtain obvious higher time and frequency resolution. Synthetic examples show that the SSWT based TF analysis can exactly capture the variable frequency components. Field data tests show the potential of the proposed approach in detecting anomalies of high-frequency attenuation and detecting the deep-layer weak signal.
Fourier finite-difference wave propagation [pdf 920K]
Xiaolei Song and Sergey Fomel
We introduce a novel technique for seismic wave extrapolation in time. The technique involves cascading a Fourier Transform operator and a finite difference operator to form a chain operator: Fourier Finite Differences (FFD). We derive the FFD operator from a pseudo-analytical solution of the acoustic wave equation. 2-D synthetic examples demonstrate that the FFD operator can have high accuracy and stability in complex velocity media. Applying the FFD method to the anisotropic case overcomes some disadvantages of other methods, such as the coupling of qP-waves and qSV-waves. The FFD method can be applied to enhance accuracy and stability of seismic imaging by reverse-time migration.
Omnidirectional plane-wave destruction [pdf 1.3M]
Zhonghuan Chen, Sergey Fomel, and Wenkai Lu
Steep structures in seismic data may bring directional aliasing, thus plane-wave destruction (PWD) filter can not obtain an accurate dip estimation. We propose to interpret plane-wave construction (PWC) filter as a line-interpolating operator and introduce a novel circle-interpolating model. The circle-interpolating PWC can avoid phase-wrapping problems, and the related circle-interpolating plane-wave destruction (PWD) can avoid aliasing problems. We design a 2D maxflat fractional delay filter to implement the circle interpolation, and prove that the 2D maxflat filter is separable in each direction. Using the maxflat fractional delay filter in the circle interpolation, we propose the omnidirectional plane-wave destruction (OPWD). The omnidirectional PWD can handle both vertical and horizontal structures. With a synthetic example, we show how to obtain an omnidirectional dip estimation using the proposed omnidirectional PWD. An application of the omnidirectional PWD to a field dataset improves the results of predictive painting and event picking, as compared to conventional PWD.
Time-to-depth conversion and seismic velocity estimation using time-migration velocity [pdf 1.0M]
Maria Cameron, Sergey Fomel, and James Sethian
The objective of this work is to build an efficient algorithm (a) to estimate seismic velocity from time-migration velocity, and (b) to convert time-migrated images to depth. We establish theoretical relations between the time-migration velocity and the seismic velocity in 2-D and 3-D using paraxial ray tracing theory. The relation in 2-D implies that the conventional Dix velocity is the ratio of the interval seismic velocity and the geometrical spreading of the image rays. We formulate an inverse problem of finding seismic velocity from the Dix velocity and develop a numerical procedure for solving it. This procedure consists of two steps: (1) computation of the geometrical spreading of the image rays and the true seismic velocity in the time-domain coordinates from the Dix velocity; (2) conversion of the true seismic velocity from the time domain to the depth domain and computation of the transition matrices from time-domain coordinates to depth. For step 1, we derive a partial differential equation (PDE) in 2-D and 3-D relating the Dix velocity and the geometrical spreading of the image rays to be found. This is a nonlinear elliptic PDE. The physical setting allows us to pose a Cauchy problem for it. This problem is ill-posed. However we are able to solve it numerically in two ways on the required interval of time. One way is a finite difference scheme inspired by the Lax-Friedrichs method. The second way is a spectral Chebyshev method. For step 2, we develop an efficient Dijkstra-like solver motivated by Sethian's Fast Marching Method. We test our numerical procedures on a synthetic data example and apply them to a field data example. We demonstrate that our algorithms give significantly more accurate estimate of the seismic velocity than the conventional Dix inversion. Our velocity estimate can be used as a reasonable first guess in building velocity models for depth imaging.
A 1-D time-varying median filter for seismic random, spike-like noise elimination [pdf 3.7M]
Yang Liu, Cai Liu, and Dian Wang
Random noise in seismic data affects the signal-to-noise ratio, obscures details, and complicates identification of useful information. We present a new method for reducing random, spike-like noise in seismic data. The method is based on a 1-D stationary median filter (MF) - the 1-D time-varying median filter (TVMF). We design a threshold value that controls the filter window according to characteristics of signal and random, spike-like noise. In view of the relationship between seismic data and the threshold value, we chose median filters with different time-varying filter windows to eliminate random, spike-like noise. When comparing our method with other common methods, e.g., the band-pass filter and stationary MF, we found that the TVMF strikes a balance between eliminating random noise and protecting useful information. To demonstrate the feasibility of our method in reducing seismic random, spike-like noise, we present results for one synthetic dataset. Results of applying the method to seismic land data from Texas demonstrate that the TVMF method is effective in practice.
Velocity-independent time-domain seismic imaging using local event slopes [pdf 1.2M]
Sergey Fomel
I show that, by estimating local event slopes in prestack seismic reflection data, it is possible to accomplish all common time-domain imaging tasks, from normal moveout to prestack time migration, without the need to estimate seismic velocities or any other attributes. Local slopes contain complete information about the reflection geometry. Once they are estimated, seismic velocities and all other moveout parameters turn into data attributes and are directly mappable from the prestack data domain into the time-migrated image domain. I develop an analytical theory for this method and demonstrate its applicability on synthetic and field data examples.
Random noise attenuation using local signal-and-noise orthogonalization [pdf 6.6M]
Yangkang Chen and Sergey Fomel
We propose a novel approach to attenuate random noise based on local signal-and-noise orthogonalization. In this approach, we first remove noise using one of the conventional denoising operators, and then apply a weighting operator to the initially denoised section in order to predict the signal-leakage energy and retrieve it from the initial noise section. The weighting operator is obtained by solving a least-squares minimization problem via shaping regularization with a smoothness constraint. Next, the initially denoised section and the retrieved signal are combined to form the final denoised section. The proposed denoising approach corresponds to orthogonalizing the initially denoised signal and noise in a local manner. We evaluate denoising performance by using local similarity. In order to test the orthogonalization property of the estimated signal and noise, we calculate the local similarity map between the denoised signal section and removed noise section. Low values of local similarity indicate a good orthogonalization and thus a good denoising performance. Synthetic and field data examples demonstrate the effectiveness of the proposed approach in applications to noise attenuation for both conventional and simultaneous-source seismic data.
Post-stack velocity analysis by separation and imaging of seismic diffractions [pdf 3.7M]
Sergey Fomel, Evgeny Landa, and M. Turhan Taner
Small geological features manifest themselves in seismic data in the form of diffracted waves, which are fundamentally different from seismic reflections. Using two field data examples and one synthetic example, we demonstrate the possibility of separating seismic diffractions in the data and imaging them with optimally chosen migration velocities. Our criterion for separating reflection and diffraction events is the smoothness and continuity of local event slopes that correspond to reflection events. For optimal focusing, we develop the local varimax measure. The objectives of this work are velocity analysis implemented in the post-stack domain and high-resolution imaging of small-scale heterogeneities. Our examples demonstrate the effectiveness of the proposed method for high-resolution imaging of such geological features as faults, channels, and salt boundaries.
Stacking seismic data using local correlation [pdf 1.4M]
Guochang Liu, Sergey Fomel, Long Jin, and Xiaohong Chen
Stacking plays an important role in improving signal-to-noise ratio and imaging quality of seismic data. However, for low-fold-coverage seismic profiles, the result of conventional stacking is not always satisfactory. To address this problem, we have developed a method of stacking in which we use local correlation as a weight for stacking common-midpoint gathers after NMO processing or common-image-point gathers after prestack migration. Application of the method to synthetic and field data showed that stacking using local correlation can be more effective in suppressing random noise and artifacts than other stacking methods.
Local skewness attribute as a seismic phase detector [pdf 904K]
Sergey Fomel and Mirko van der Baan
We propose a novel seismic attribute, local skewness, as an indicator of localized phase of seismic signals. The proposed attribute appears to have a higher dynamical range and a better stability than the previously used local kurtosis. Synthetic and real data examples demonstrate the effectiveness of local skewness in detecting and correcting time-varying, locally-observed phase of seismic signals.
A fast algorithm for 3D azimuthally anisotropic velocity scan [pdf 1.4M]
Jingwei Hu, Sergey Fomel, and Lexing Ying
Conventional velocity scan can be computationally expensive for large-size seismic data, particularly when the presence of anisotropy requires multiparameter estimation. We introduce a fast algorithm for 3D azimuthally anisotropic velocity scan, which is a generalization of the previously proposed 2D butterfly algorithm for hyperbolic Radon transform. To compute the semblance in a two-parameter residual moveout domain, the numerical complexity of our algorithm is roughly $O(N^3N)$ as opposed to $O(N^5)$ of the straightforward velocity scan, with $N$ being representative of the number of points in either dimension of data space or parameter space. We provide both synthetic and field-data examples to illustrate the efficiency and accuracy of the algorithm.
Stacking angle-domain common-image gathers for normalization of illumination [pdf 1.2M]
Guochang Liu, Sergey Fomel, and Xiaohong Chen
Unequal illumination of the subsurface highly impacts the quality of seismic imaging. Different image points of the media have different folds of reflection-angle illumination, which can be caused by irregular acquisition or by wave propagation in complex media. To address this problem, we present a method of stacking angle-domain common-image gathers (ADCIGs), in which we use local similarity with soft thresholding to decide the folds of local illumination. Normalization by local similarity regularizes local illumination of reflection angles for each image point of the subsurface model. This approach can restore good fidelity of amplitude by selective stacking in the image space, whatever the cause of acquisition or propagation irregularities. We use two synthetic examples to demonstrate that our method can normalize migration amplitudes and effectively suppress migration artifacts.
Automated spectral recomposition with application in stratigraphic interpretation [pdf 2.3M]
Yihua Cai, Sergey Fomel, and Hongliu Zeng
Analyzing seismic attributes in the frequency domain is helpful for reservoir characterization. To analyze the reservoir interval of interest in detail, it is important to capture the seismic response at each frequency subset. Spectral recomposition can be used to extract significant components from the seismic spectrum. We propose a separable nonlinear least-squares algorithm for spectral recomposition, which estimates both linear and nonlinear parts automatically in separate steps. Our approach is applied to estimate fundamental signal parameters, peak frequencies and amplitudes, with which the seismic spectrum can be reconstructed. Automated spectral recomposition helps us visualize frequency-dependent geological features on both cross sections and time slices by extracting significant frequency components. Spectral recomposition can also indicate how frequency contents attenuate with time.
Nonlinear structure-enhancing filtering using plane-wave prediction [pdf 3.7M]
Yang Liu, Sergey Fomel, and Guochang Liu
Attenuation of random noise and enhancement of structural continuity can significantly improve the quality of seismic interpretation. We present a new technique, which aims at reducing random noise while protecting structural information. The technique is based on combining structure prediction with either similarity-mean filtering or lower-upper-middle (LUM) filtering. We use structure prediction to form a structural prediction of seismic traces from neighboring traces. We apply a nonlinear similarity-mean filter or an LUM filter to select best samples from different predictions. In comparison with other common filters, such as mean or median, the additional parameters of the nonlinear filters allow us to better control the balance between eliminating random noise and protecting structural information. Numerical tests using synthetic and field data show the effectiveness of the proposed structure-enhancing filters.
A parallel sweeping preconditioner for heterogeneous 3D Helmholtz equations [pdf 500K]
Jack Poulson, Bj"orn Engquist, Siwei Li, and Lexing Ying
A parallelization of a sweeping preconditioner for 3D Helmholtz equations without large cavities is introduced and benchmarked for several challenging velocity models. The setup and application costs of the sequential preconditioner are shown to be $O(^2 N^{4/3})$ and $O(N N)$ , where $()$ denotes the modestly frequency-dependent number of grid points per Perfectly Matched Layer. Several computational and memory improvements are introduced relative to using black-box sparse-direct solvers for the auxiliary problems, and competitive runtimes and iteration counts are reported for high-frequency problems distributed over thousands of cores. Two open-source packages are released along with this paper: Parallel Sweeping Preconditioner (PSP) and the underlying distributed multifrontal solver, Clique.
Modeling of pseudo-acoustic P-waves in orthorhombic media with a lowrank approximation [pdf 808K]
Xiaolei Song and Tariq Alkhalifah
Wavefield extrapolation in pseudo-acoustic orthorhombic anisotropic media suffers from wave-mode coupling and stability limitations in the parameter range. We use the dispersion relation for scalar wave propagation in pseudo-acoustic orthorhombic media to model acoustic wavefields. The wavenumber-domain application of the Laplacian operator allows us to propagate the P-waves exclusively, without imposing any conditions on the parameter range of stability. It also allows us to avoid dispersion artifacts commonly associated with evaluating the Laplacian operator in space domain using practical finite difference stencils. To handle the corresponding space-wavenumber mixed-domain operator, we apply the lowrank approximation approach. Considering the number of parameters necessary to describe orthorhombic anisotropy, the lowrank approach yields space-wavenumber decomposition of the extrapolator operator that is dependent on space location regardless of the parameters, a feature necessary for orthorhombic anisotropy. Numerical experiments show that the proposed wavefield extrapolator is accurate and practically free of dispersion. Furthermore, there is no coupling of qSv and qP waves, because we use the analytical dispersion solution corresponding to the $P$-wave.
A robust approach to time-to-depth conversion and interval velocity estimation from time migration in the presence of lateral velocity variations [pdf 2.0M]
Siwei Li and Sergey Fomel
The problem of conversion from time-migration velocity to an interval velocity in depth in the presence of lateral velocity variations can be reduced to solving a system of partial differential equations. In this paper, we formulate the problem as a nonlinear least-squares optimization for seismic interval velocity and seek its solution iteratively. The input for inversion is the Dix velocity which also serves as an initial guess. The inversion gradually updates the interval velocity in order to account for lateral velocity variations that are neglected in the Dix inversion. The algorithm has a moderate cost thanks to regularization that speeds up convergence while ensuring a smooth output. The proposed method should be numerically robust compared to the previous approaches, which amount to extrapolation in depth monotonically. For a successful time-to-depth conversion, image-ray caustics should be either nonexistent or excluded from the computational domain. The resulting velocity can be used in subsequent depth-imaging model building. Both synthetic and field data examples demonstrate the applicability of the proposed approach.
Predictive painting of 3-D seismic volumes [pdf 848K]
Sergey Fomel
Structural information is the most important content of seismic images. I introduce a numerical algorithm for spreading information in 3-D volumes according to the local structure of seismic events. The algorithm consists of two steps. First, local spatially-variable inline and crossline slopes of seismic events are estimated by the plane-wave-destruction method. Next, a seed trace is inserted in the volume, and the information contained in that trace is spread inside the volume, thus automatically ``painting'' the data space. Immediate applications of this technique include automatic horizon picking and flattening in applications to both prestack and post-stack seismic data analysis. Synthetic and field data tests demonstrate the effectiveness of predictive painting.
Seislet transform and seislet frame [pdf 3.0M]
Sergey Fomel and Yang Liu
We introduce a digital wavelet-like transform, which is tailored specifically for representing seismic data. The transform provides a multiscale orthogonal basis with basis functions aligned along seismic events in the input data. It is defined with the help of the wavelet lifting scheme combined with local plane-wave destruction. In the 1-D case, the seislet transform is designed to follow locally sinusoidal components. In the 2-D case, it is designed to follow local plane wave components with smoothly variable slopes. If more than one component is present, the transform turns into an overcomplete representation or a tight frame. In these terms, the classic digital wavelet transform is simply a seislet transform for a zero frequency (in 1-D) or zero slope (in 2-D). The main objective of the new transform is an effective seismic data compression for designing efficient data analysis algorithms. Traditional signal processing tasks such as noise attenuation and trace interpolation become simply defined in the seislet domain. When applied in the offset direction on common midpoint or common image point gathers, the seislet transform finds an additional application in optimal stacking of seismic records.
3D velocity-independent elliptically-anisotropic moveout correction [pdf 1.3M]
William Burnett and Sergey Fomel
Azimuthal anisotropy or lateral velocity variations cause azimuthal variations in moveout velocity which can lead to seismic image degradation if not properly handled. In cases where apparent azimuthally anisotropic moveout is present, a single picked velocity is inadequate to flatten an event on a 3D CMP gather. Conventional velocity analysis techniques require a significant amount of time and effort, especially in areas where apparent anisotropy is observed. We propose a velocity-independent imaging approach to perform an elliptically anisotropic moveout correction in 3D. The velocity-independent approach relies on volumetric local traveltime slopes rather than aggregate velocities, and therefore provides an azimuthally flexible description of traveltime geometries throughout the gather. We derive theoretical expressions for extracting the moveout slowness matrix and the angle between the symmetry and acquisition axes as volumetric local attributes. A practical inversion scheme to extract the same parameters is also developed. These parameters are used to solve for moveout slowness as a function of azimuth. Tests on a synthetic CMP gather show accurate results for the automatic moveout correction and the inversion scheme. A field data example from West Texas illustrates the application of the automatic moveout correction as a residual moveout.
Structural uncertainty of time-migrated seismic images [pdf 728K]
Sergey Fomel and Evgeny Landa
Structural information in seismic images is uncertain. The main cause of this uncertainty is uncertainty in velocity estimation. We adopt the technique of velocity continuation for estimating velocity uncertainties and corresponding structural uncertainties in time-migrated images. Data experiments indicate that structural uncertainties can be significant even when both structure and velocity variations are mild.
Time-lapse image registration using the local similarity attribute [pdf 480K]
Sergey Fomel and Long Jin
We present a method for registration of time-lapse seismic images based on the local similarity attribute. We define registration as an automatic point-by-point alignment of time-lapse images. Stretching and squeezing a monitor image and computing its local similarity to the base image allows us to detect an optimal registration even in the presence of significant velocity changes in the overburden. A by-product of this process is an estimate of the ratio of the interval seismic velocities in the reservoir interval. We illustrate the proposed method and demonstrate its effectiveness using both synthetic experiments and real data from the Duri time-lapse experiment in Indonesia.
Fractal heterogeneities in sonic logs and low-frequency scattering attenuation [pdf 2.3M]
Thomas J. Browaeys and Sergey Fomel
Cycles in sedimentary strata exist at different scales and can be described by fractal statistics. We use von Kármán's autocorrelation function to model heterogeneities in sonic logs from a clastic reservoir and propose a nonlinear parameter estimation. Our method is validated using synthetic signals, and when applied to real sonic logs, it extracts both the fractal properties of high spatial frequencies and one dominant cycle between 2.5 and 7 m. Results demonstrate non-Gaussian and antipersistent statistics of sedimentary layers. We derive an analytical formula for the scattering attenuation of scalar waves by 3D isotropic fractal heterogeneities using the mean field theory. Penetration of waves exhibits a high-frequency cutoff sensitive to heterogeneity size. Therefore shear waves can be more attenuated than compressional waves because of their shorter wavelength.
Adaptive multiple subtraction using regularized nonstationary regression [pdf 1.4M]
Sergey Fomel
Stationary regression is the backbone of different seismic data processing algorithms including match filtering, which is commonly applied for adaptive multiple subtraction. However, the assumption of stationarity is not always adequate for describing seismic signals. I present a general method of nonstationary regression and show its application to nonstationary match filtering. The key idea is the use of shaping regularization for constraining the variability of nonstationary regression coefficients. As shown by simple computational experiments, shaping regularization has clear advantages over conventional Tikhonov's regularization, incuding a more intuitive selection of parameters and a faster iterative convergence. Using benchmark synthetic data examples, I demonstrate successful applications of this method to the problem of adaptive subtraction of multiple reflections.
First-break traveltime tomography with the double-square-root eikonal equation [pdf 2.5M]
Siwei Li, Alexander Vladimirsky, and Sergey Fomel
First-break traveltime tomography is based on the eikonal equation. Since the eikonal equation is solved at fixed shot positions and only receiver positions can move along the ray-path, the adjoint-state tomography relies on inversion to resolve possible contradicting information between independent shots. The double-square-root eikonal equation allows not only the receivers but also the shots to change position, and thus describes the prestack survey as a whole. Consequently, its linearized tomographic operator naturally handles all shots together, in contrast with the shot-wise approach in the traditional eikonal-based framework. The double-square-root eikonal equation is singular for the horizontal waves, which require special handling. Although it is possible to recover all branches of the solution through post-processing, our current forward modeling and tomography focus on the diving wave branch only. We consider two upwind discretizations of the double-square-root eikonal equation and show that the explicit scheme is only conditionally convergent and relies on non-physical stability conditions. We then prove that an implicit upwind discretization is unconditionally convergent and monotonically causal. The latter property makes it possible to introduce a modified fast marching method thus obtaining first-break traveltimes both efficiently and accurately. To compare the new double-square-root eikonal-based tomography and traditional eikonal-based tomography, we perform linearizations and apply the same adjoint-state formulation and upwind finite-differences implementation to both approaches. Synthetic model examples justify that the proposed approach converges faster and is more robust than the traditional one.
Generalized nonhyperbolic moveout approximation [pdf 792K]
Sergey Fomel and Alexey Stovas
Reflection moveout approximations are commonly used for velocity analysis, stacking, and time migration. We introduce a novel functional form for approximating the moveout of reflection traveltimes at large offsets. While the classic hyperbolic approximation uses only two parameters (the zero-offset time and the moveout velocity), our form involves five parameters, which can be determined, in a known medium, from zero-offset computations and from tracing one non-zero-offset ray. We call it a generalized approximation because it reduces to some known three-parameter forms (the shifted hyperbola of Malovichko, de Baziliere, and Castle; the Padé approximation of Alkhalifah and Tsvankin; and others) with a particular choice of coefficients. By testing the accuracy of the proposed approximation with analytical and numerical examples, we show that it can bring several-orders-of-magnitude improvement in accuracy at large offsets compared to known analytical approximations, which makes it as good as exact for many practical purposes.
Lowrank seismic wave extrapolation on a staggered grid [pdf 872K]
Gang Fang, Sergey Fomel, Qizhen Du, and Jingwei Hu
We propose a new spectral method and a new finite-difference method for seismic wave extrapolation in time. Using staggered temporal and spatial grids, we derive a wave extrapolation operator using a lowrank decomposition for a first-order system of wave equations and design the corresponding finite-difference scheme. The proposed methods extend previously proposed lowrank and lowrank finite-difference wave extrapolation methods from the cases of constant density to those of variable density. Dispersion analysis demonstrates that the proposed methods have high accuracy for a wide wavenumber range and significantly reduce the numerical dispersion. The method of manufactured solutions coupled with mesh refinement is used to verify each method and to compare numerical errors. 2-D synthetic examples demonstrate that the proposed method is highly accurate and stable. The proposed methods can be used for seismic modeling or reverse time migration.
Deblending using normal moveout and median filtering in common-midpoint gathers [pdf 5.6M]
Yangkang Chen, Jiang Yuan, Zhaoyu Jin, Keling Chen, and Lele Zhang
The benefits of simultaneous source acquisition are compromised by the challenges of dealing with intense blending noise. In this paper, we propose a processing workflow for blended data. The incoherent property of blending noise in the common-midpoint (CMP) gathers is utilized for applying median filtering along the spatial direction after normal moveout (NMO) correction. The key step in the proposed workflow is that we need to obtain a precise velocity estimation which is required by the subsequent NMO correction. Because of the intense blending noise, the velocity scan can not be obtained in one step. We can recursively polish both deblended result and velocity estimation by deblending using the updated velocity estimation and velocity scanning using the updated deblended result. We use synthetic and field data examples to demonstrate the performance of the proposed approach. The migrated image of deblended data is cleaner than that of blended data, and is similar to that of unblended data.
Azimuthally anisotropic 3D velocity continuation [pdf 1.2M]
William Burnett and Sergey Fomel
We extend time-domain velocity continuation to the zero-offset 3D azimuthally anisotropic case. Velocity continuation describes how a seismic image changes given a change in migration velocity. This description turns out to be of a wave propagation process, in which images change along a velocity axis. In the anisotropic case, the velocity model is multi-parameter. Therefore, anisotropic image propagation is multi-dimensional. We use a three-parameter slowness model, which is related to azimuthal variations in velocity, as well as their principal directions. This information is useful for fracture and reservoir characterization from seismic data. We provide synthetic diffraction imaging examples to illustrate the concept and potential applications of azimuthal velocity continuation and to analyze the impulse response of the 3D velocity continuation operator.
Time-frequency analysis of seismic data using local attributes [pdf 1.8M]
Guochang Liu, Sergey Fomel, and Xiaohong Chen
Time-frequency analysis is an important technology in seismic data processing and interpretation. To localize frequency content in time, we have developed a novel method for computing a time-frequency map for nonstationary signals using an iterative inversion framework. We calculated time-varying Fourier coefficients by solving a least-squares problem that uses regularized nonstationary regression. We defined the time-frequency map as the norm of time-varying coefficients. Time-varying average frequency of the seismic data can also be estimated from the time-frequency map calculated by our method. We tested the method on benchmark synthetic signals and compared it with the well-known Stransform. Two field data examples showed applications of the proposed method for delineation of sand channels and for detection of low-frequency anomalies.
Velocity analysis using $AB$ semblance [pdf 1.2M]
Sergey Fomel
I derive and analyze an explicit formula for a generalized semblance attribute, which is suitable for velocity analysis of prestack seismic gathers with distinct amplitude trends. While the conventional semblance can be interpreted as squared correlation with a constant, the $AB$ semblance is defined as a correlation with a trend. This measure is particularly attractive for analyzing class II AVO anomalies and converted waves. Analytical derivations and numerical experiments show that the resolution of the $AB$ semblance is approximately twice lower than that of the conventional semblance. However, this does not prevent it from being an effective attribute. I use synthetic and field data examples to demonstrate the improvements in velocity analysis from $AB$ semblance.
Non-hyperbolic common reflection surface [pdf 1.2M]
Sergey Fomel and Roman Kazinnik
The method of common reflection surface (CRS) extend conventional stacking of seismic traces over offset to multidimensional stacking over offset-midpoint surfaces. We propose a new form of the stacking surface, derived from the analytical solution for reflection traveltime from a hyperbolic reflector. Both analytical comparisons and numerical tests show that the new approximation can be significantly more accurate than the conventional CRS approximation at large offsets or at large midpoint separations while using essentially the same parameters.
Carbonate reservoir characterization using seismic diffraction imaging [pdf 4.4M]
Luke Decker, Xavier Janson, and Sergey Fomel
Although extremely prolific worldwide, carbonate reservoirs are challenging to characterize using traditional seismic reflection imaging techniques. We use computational experiments with synthetic models to demonstrate the possibility seismic diffraction imaging has of overcoming common obstacles associated with seismic reflection imaging and aiding interpreters of carbonate systems. Diffraction imaging improves the horizontal resolution of individual voids in a karst reservoir model and identification of heterogeneous regions below the resolution of reflections in a reservoir scale model.
Random noise attenuation by $f$-$x$ empirical mode decomposition predictive filtering [pdf 14M]
Yangkang Chen and Jitao Ma
Random noise attenuation always plays an important role in seismic data processing. One of the most widely used methods for suppressing random noise is $f-x$ predictive filtering. When the subsurface structure becomes complex, this method suffers from higher prediction errors owing to the large number of different dip components that need to be predicted. In this paper, we propose a novel denoising method termed $f-x$ empirical mode decomposition predictive filtering (EMDPF). This new scheme solves the problem that makes $f-x$ empirical mode decomposition (EMD) ineffective with complex seismic data. Also, by making the prediction more precise, the new scheme removes the limitation of conventional $f-x$ predictive filtering when dealing with multi-dip seismic profiles. In this new method, we first apply EMD to each frequency slice in the $f-x$ domain and obtain several intrinsic mode functions (IMF). Then an auto-regressive (AR) model is applied to the sum of the first few IMFs, which contain the high-dip-angle components, in order to predict the useful steeper events. Finally, the predicted events are added to the sum of the remaining IMFs. This process improves the prediction precision by utilizing an EMD based dip filter to reduce the dip components before $f-x$ predictive filtering. Both synthetic and real data sets demonstrate the performance of our proposed method in preserving more useful energy.

TCCS -- TABLE OF CONTENTS