JSG -- TABLE OF CONTENTS

JSG -- TABLE OF CONTENTS

Stacking angle-domain common-image gathers for normalization of illumination [pdf 1.9M]
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.
Seislet transform and seislet frame [pdf 3.5M]
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.
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.
Fractal heterogeneities in sonic logs and low-frequency scattering attenuation [pdf 2.4M]
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 1.8M]
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.
Velocity analysis using $AB$ semblance [pdf 1.8M]
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.
Predictive painting of 3-D seismic volumes [pdf 1.6M]
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.
Stacking seismic data using local correlation [pdf 1.9M]
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.
Time-lapse image registration using the local similarity attribute [pdf 740K]
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.
Seismic wave extrapolation using lowrank symbol approximation [pdf 2.3M]
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.
Shaping regularization in geophysical estimation problems [pdf 1.6M]
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.
Azimuthally anisotropic 3D velocity continuation [pdf 2.0M]
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.
Accelerated plane-wave destruction [pdf 2.0M]
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.
Post-stack velocity analysis by separation and imaging of seismic diffractions [pdf 6.6M]
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.
A reversible transform for seismic data processing [pdf 896K]
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.
Local seismic attributes [pdf 208K]
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.
Velocity-independent time-domain seismic imaging using local event slopes [pdf 2.2M]
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.
OC-seislet: seislet transform construction with differential offset continuation [pdf 5.8M]
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.
Seismic data analysis using local time-frequency decomposition [pdf 2.8M]
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.
Fourier finite-difference wave propagation [pdf 1.3M]
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.
Time-frequency analysis of seismic data using local attributes [pdf 3.1M]
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.
Time-to-depth conversion and seismic velocity estimation using time-migration velocity [pdf 1.6M]
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.
Non-hyperbolic common reflection surface [pdf 1.4M]
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.
Theory of 3-D angle gathers in wave-equation seismic imaging [pdf 468K]
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.
Lowrank finite-differences and lowrank Fourier finite-differences for seismic wave extrapolation in the acoustic approximation [pdf 1.4M]
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.
Nonlinear structure-enhancing filtering using plane-wave prediction [pdf 6.6M]
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.
Generalized nonhyperbolic moveout approximation [pdf 956K]
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.
3D velocity-independent elliptically-anisotropic moveout correction [pdf 2.2M]
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.
Seismic data interpolation beyond aliasing using regularized nonstationary autoregression [pdf 2.2M]
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.
On anelliptic approximations for $qP$ velocities in transversally isotropic media [pdf 528K]
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.

JSG -- TABLE OF CONTENTS