SEP-114 (2003)
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Imaging
Seismic imaging using Riemannian wavefield extrapolation
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Sava P. and Fomel S.
Abstract
Riemannian spaces are described by non-orthogonal curvilinear coordinates. We generalize one-way wavefield extrapolation to semi-orthogonal Riemannian coordinate systems, which include, but are not limited to, ray coordinate systems. We obtain one-way wavefield extrapolation methods which are not dip-limited, and which can even be used to image overturning waves. Ray coordinate systems can be initiated either from point sources, or from plane waves incident at various angles. Since wavefield propagation happens mostly along the extrapolation direction, we can use cheap finite-difference or mixed-domain extrapolators to achieve high angle accuracy. The main applications of our method include imaging of steeply dipping or overturning reflections.
Wavefield extrapolation in phase-ray coordinates
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Shragge J. and Biondi B.
Abstract
A ray theoretic formulation is developed that allows rays to be traced directly from existing solutions to the Helmholtz equation. These rays, termed phase-rays, are defined by the direction normal to surfaces of constant wavefield phase. Phase-rays exhibit a number of attractive characteristics, including triplication-free ray-fields, an ability to shoot rays forward or backward, and an ability to shoot infill rays for ensuring adequate ray density. Because of these traits, we use phase-rays as a coordinate basis on which to extrapolate wavefields using the generalized coordinate system approach. Examples of wavefields successfully extrapolated in phase-ray coordinates are presented, and the merits and drawbacks of this approach, relative to conventionally traced ray coordinates, are discussed.
Amplitude balancing of 3-D angle-domain common-image gathers
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Biondi B.
Abstract
The azimuthal resolution of 3-D Angle Domain Common Image Gathers (ADCIGs) strongly varies with the reflection aperture angle. This dependence may cause severe distortions in the image when 3-D ADCIGs are averaged over azimuths. To correct for these distortions, I derive an effective weighting method based on the jacobian of the transformation to angle domain. The proposed method avoids the underweighting of the reflections close to normal incidence by properly taking into account the ``folding'' of the azimuth axis. A simple scheme to limit the range of the azimuthal averaging as a function of the opening angle further attenuates noise in the image. A synthetic example illustrates the practical application of the proposed methodology .
Velocity sensitivity of subsalt imaging through regularized inversion
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Clapp M. L.
Abstract
The effects of inaccurate velocity models on migration are well known. Accurate velocity models are most difficult to obtain in complex areas where iterative inversion can provide a better image than migration. This paper investigates the velocity sensitivity of a regularized inversion scheme that explicitly assumes that the correct velocity is being used. This inversion uses a regularization operator that assumes that there is no moveout along the offset ray parameter axis. Experiments performed with various incorrect velocity models indicate that this assumption is valid for velocity models that can be reasonably produced by common velocity analysis techniques. Velocity models that are very inaccurate cause the inversion process to reject attempts by the regularization to produce an image that is inconsistent with the data.
WKBJ and amplitude preserving one-way wave equation
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Shan G. and Biondi B.
Abstract
The standard one-way wave equation Claerbout (1971) produces the correct phase of the wavefield, but it is not equivalent to the acoustic wave equation in terms of amplitude. Zhang (1993) suggests that to improve the dynamics information of one-way wave equation, an amplitude correction term should be included into the standard one-way wave equation. Zhang et al. (2001) apply the one-way wave equation with the amplitude correction to shot-profile migration and shows that it can provide the same amplitude as an image produced by the true amplitude Kirchoff migration Hanitzsch (1997) by asymptotic analysis. ...
Kinematics of 3-D angle-domain common-image gathers for migration velocity analysis
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Tisserant T. and Biondi B.
Abstract
We extend the study of the kinematic properties of 2-D offset- and angle-domain common-image gathers to the general 3-D problem. We examine how the use of an incorrect migration velocity affects the behavior of the image in the offset and angle domains. We show that the general 3-D case with and without the correct migration velocity can be cast into a 2-D formulation, making it possible to apply existing theory from 2-D. We illustrate both ray-tracing and plane-wave approaches to the problem and verify our theoretical results with a synthetic model.
Velocity Analysis
Wave-equation MVA: Born, Rytov and beyond
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Sava P. and Biondi B.
Abstract
The linearized wave-equation MVA operator can be used for velocity analysis using both Born and Rytov approximations. The distinction arises from the method used to compute the image perturbations. Both approximations suffer from limitations that limit their practicality: the Born approximation is usable only for small anomalies, while the Rytov approximation requires phase unwrapping. Differential image perturbations can be used for arbitrarily large slowness anomalies and do not require phase unwrapping, but their accuracy decreases with increasing deviation from the background image. For simple cases, the differential image perturbation method is equivalent with phase-unwrapped Rytov.
Dip-dependent residual moveout
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Shan G. and Biondi B.
Abstract
Residual moveout is an effective tool for interval velocity analysis for depth migration. However, the conventional residual moveout method assumes that the reflectors in the subsurface are all flat, and thus estimates curvature parameters for velocity analysis that are inaccurate for steeply dipping reflectors. In this paper, we develop a dip-dependent residual moveout method, which is performed in the Fourier domain and handles dip effects.
Velocity estimation for seismic data exhibiting focusing-effect AVO (Part 3)
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Vlad I., Biondi B., and Sava P.
Abstract
Given a satisfactory image perturbation, Target Image Fitting (TIF) Wave Equation Migration Velocity Analysis (WEMVA) successfully produces a velocity model that completely eliminates focusing-effect AVO anomalies through prestack depth migration. However, TIF WEMVA ceases to converge to the desired result when the Born approximation is not fulfilled because the starting guess is too far from the correct velocity. Fortunately, there are several possible remedies to this problem. Extracting the image perturbation proves somewhat more complicated than expected, but new possible solutions based on differential semblance have been identified.
Reflection tomography with depth control
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Chen W., Clapp R. G., and Biondi B.
Abstract
Reflection tomography Clapp (2001); Stork and Clayton (1991); Stork (1992) is one of the most effective and widely used velocity estimation methods. However, reflection tomography has velocity-depth ambiguity problem (we do not know how much a traveltime error is due to a velocity error and how much is due to a reflector misposition) because of insufficient source-receiver offset and lateral velocity changes Bickel (1990); Lines (1993); Ross (1994); Tieman (1994). From borehole data, we can obtain the correct reflection positions around the borehole. The normal shift between the correct reflection positions and the apparent reflection positions can be linearly mapped to the traveltime perturbation along the normal ray van Trier (1990). However, from borehole data, we can only obtain the correct position for only a few reflection points along the borehole. In this paper, we assume all the reflection points within a local area around the borehole have the same normal shift. The normal ray traveltime perturbation for all these reflection points are then backpropagated simultaneously with the reflection traveltime perturbation. We applied this scheme on a synthetic model and obtained a better inversion result than using reflection tomography without this control. We further discuss how to improve this method for more complex datasets. ...
Frequency-dependent velocity analysis?
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Vlad I.
Abstract
There are two reasons for exploring imaging and velocity analysis with a frequency-dependent velocity model. First, significant dispersion may occur naturally in the volume of rock investigated by the seismic experiment. Second, the goal of Wave Equation Migration Velocity Analysis (WEMVA, Biondi and Sava (1999)) is image optimization, which some successful approaches Pratt (1999) perform by inverting one frequency at a time, from lower to higher frequencies. The velocity model that optimizes the ...
Novel inversion methods
Interval velocity estimation using edge-preserving regularization
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Valenciano A. A., Brown M., Sacchi M. D., and Guitton A.
Abstract
We test two edge-preserving forms of model regularization in a least-squares implementation of Dix formula that leads to interval velocities with sharp edges in the plane. This characteristic in the interval velocity may be desirable in geologic environments with abrupt changes in velocity, like: carbonate layers, salt bodies, and strong faulting.
Basis pursuit for geophysical inversion
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Artman B. and Sacchi M.
Abstract
Accepting an inversion principle, it is possible to design an algorithm to meet any requirements or constraints. Given the context of representing a signal with an arbitrary overcomplete dictionary of waveforms within the signal, one can design an inversion algorithm that will focus energy into a small number of model space coefficients. With this principle in mind, an analogy to linear programming is developed that results in an algorithm with: the properties of an l1 norm, a small and stable parameter space, definable convergence, and impressive denoising capabilities. Linear programming methods solve a nonlinear problem in an interior/barrier loop method similar to iteratively reweighted least squares (IRLS) algorithms, and are much slower than a least squares solution obtained with a conjugate gradient method. Velocity scanning with the hyperbolic radon transform is implemented as a test case for the methodology.
Deblurring using edge-preserving regularization
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Valenciano A. A. and Brown M.
Abstract
In this short note, we test various edge-preserving regularization schemes in the context of deblurring a text image with random noise. The blurred text image was created by Nagy and O'Leary (2003a) as a test case. Even if the blurring filter is known exactly, as it is in this case, sharp features are nearly in the nullspace of the filter which we must ``invert'', or deblur. Those eigenvalues of the filter matrix corresponding to edges may be well below the noise level, and thus difficult or impossible to resolve. We know that letters should be homogeneous for intervals (piecewise constant), thus it makes sense to impose model smoothness using a regularization operator. But letters also have abrupt discontinuities, thus using a regularization operator that imposes model smoothness considerably degrades our ability to discern the letter edges. ...
Irregular data
AMO regularization: Effective approximate inverses for amplitude preservation
(ps.gz 319K) (pdf 396K) (src 96772K)
Clapp R. G.
Abstract
Most downward continuation methods require that the data lie on a regular mesh. To map the irregular recorded seismic data onto the regular mesh is a far from trivial exercise. A common approach in industry is to think of the problems in the same way we approach Kirchhoff migration, namely to loop over data space and spread into our regular model space. The spreading operation is governed by something like AMO Biondi et al. (1998), which maps data from one offset vector to ...
More fitting equations for PEF estimation on sparse data
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Curry W.
Abstract
Prediction-error filters (PEFs) can be used to interpolate data Claerbout (1999); Crawley (2000); Spitz (1991). In order to generate a PEF, a least-squares problem is solved where regularly-sampled data is convolved with an unknown PEF. When estimating a PEF with missing data, the equations that contain missing data can be eliminated from the inversion. When dealing with sparse data, however, all equations may contain missing data, so ...
Dip estimation from irregularly-sampled seismic data
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Brown M.
Abstract
The geometry of reflection seismic experiments rarely conforms to an idealized, or nominal geometry. This is especially true in regions with unavoidable surface obstructions like rivers, towns, or existing offshore oil platforms. Such deviations from nominal geometry render many computer seismic processing applications ineffective. It is therefore of considerable importance to 1) accurately map irregularly-sampled seismic data onto a regular grid with sufficient spatial resolution, and 2) to interpolate any missing trace locations with reasonable values. Any interpolation method fills missing traces using a prior estimate of the ...
Prediction-error filter estimation on irregular traces
(ps.gz 39K) (pdf 103K) (src 73K)
Curry W.
Abstract
Regularly-sampled data is normally required to estimate a prediction-error filter (PEF). I show how a small PEF can be estimated when the data is irregularly sampled in one dimension. I then estimate a PEF on an irregularly-sampled 2D examples, and discuss extension to 3D.
Image Segmentation
Image segmentation for tracking salt boundaries
(ps.gz 83K) (pdf 179K) (src 178K)
Lomask J. and Biondi B.
Abstract
Image segmentation can be used to track salt boundaries when the salt boundary amplitude is greater than any other local reflections. We apply a modified version of the normalized cut image segmentation method to partition seismic images along salt boundaries. In principle our method should work even when the boundaries are not continuous, and conventional horizon tracking algorithms may fail. Our implementation of this method calculates a weight connecting each pixel in the image to each pixel in a local neighborhood. The magnitude of the weight is inversely proportional to the size of the maximum instantaneous amplitude along the shortest path between the two pixels. This method is demonstrated to be effective on two simple models and 2D seismic section.
Automatic discontinuity extraction for 3-D seismic images
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Ji J.
Abstract
A goal of seismic processing is to verify the spatial characteristics of the subsurface. Achieving this goal often requires human analysis to interpret geologically meaningful discontinuities from the seismic image. This interpretation is a challenging task even for an experienced interpreter if the image is 3D. This paper introduces an automatic method to extract seismic event discontinuities from a 3D seismic image. The proposed method consists of three steps. The first step is to evaluate the coherency of seismic events from the seismic image. The second step is to express the regions where discontinuities may exist in a binary image form. The third step is to locate the discontinuity surfaces by thinning the region found in the second step.
Rock Physics
Elastic and poroelastic analysis of Thomsen parameters for seismic waves in finely layered VTI media
(ps.gz 67K) (pdf 179K) (src 64K)
Berryman J. G.
Abstract
Layered earth models are well justified by experience, and provide a simple means of studying fairly general behavior of the elastic and poroelastic characteristics of seismic waves in the earth. Thomsen's anisotropy parameters for weak elastic and poroelastic anisotropy are now commonly used in exploration, and can be conveniently expressed in terms of the layer averages of Backus. Since our main interest is usually in the fluids underground, it would be helpful to have a set of general equations relating the Thomsen parameters as directly as possible to the fluid properties. This end can be achieved in a rather straightforward fashion for these layered earth models, and the present paper develops and then discusses these relations. It is found that, although there are five effective shear moduli for any layered VTI medium, one and only one effective shear modulus for the layered system contains all the dependence of pore fluids on the elastic or poroelastic constants that can be observed in vertically polarized shear waves in VTI media. The effects of the pore fluids on this effective shear modulus can be substantial (as much as a factor of 5 in the examples presented here) when the medium behaves in an undrained fashion, as might be expected at higher frequencies such as sonic and ultrasonic waves for well-logging or laboratory experiments, or at seismic wave frequencies for low permeability regions of reservoirs, prior to hydrofracing. The results presented are strictly for velocity analysis, not for amplitude or attenuation.
Scale-up in poroelastic systems and applications to reservoirs
(ps.gz 43K) (pdf 116K) (src 17K)
Berryman J. G.
Abstract
A fundamental problem of heterogeneous systems is that the macroscale behavior is not necessarily well-described by equations familiar to us at the meso- or microscale. In relatively simple cases like electrical conduction and elasticity, it is true that the equations describing macroscale behavior take the same form as those at the microscale. But in more complex systems, these simple results do not hold. Consider fluid flow in porous media where the microscale behavior is well-described by Navier-Stokes' equations for liquid in the pores while the macroscale behavior instead obeys Darcy's equation. Rigorous methods for establishing the form of such equations for macroscale behavior include multiscale homogenization methods and also the volume averaging method. In addition, it has been shown that Biot's equations of poroelasticity follow in a scale-up of the microscale equations of elasticity coupled to Navier-Stokes. Laboratory measurements have shown that Biot's equations indeed hold for simple systems but heterogeneous systems can have quite different behavior. So the question arises whether there is yet another level of scale-up needed to arrive at equations valid for the reservoir scale? And if so, do these equations take the form of Biot's equations or some other form? We will discuss these issues and show that the double-porosity equations play a special role in the scale-up to equations describing reservoir behavior, for fluid pumping, geomechanics, as well as seismic wave propagation.
Seismic attenuation due to wave-induced flow
(ps.gz 221K) (pdf 356K) (src 352K)
Pride S. R., Berryman J. G., and Harris J. M.
Abstract
Analytical expressions for three P-wave attenuation mechanisms in sedimentary rocks are given a unified theoretical framework. Two of the models concern wave-induced flow due to heterogeneity in the elastic moduli at ``mesoscopic'' scales (scales greater than grain sizes but smaller than wavelengths). In the first model, the heterogeneity is due to lithological variations (e.g., mixtures of sands and shales) with a single fluid saturating all the pores. In the second model, a single uniform lithology is saturated in mesoscopic ``patches'' by two immiscible fluids (e.g., air and water). In the third model, the heterogeneity is at ``microscopic'' grain scales (broken grain contacts and/or micro-cracks in the grains) and the associated fluid response corresponds to ``squirt flow.'' The model of squirt flow derived here reduces to proper limits as any of the fluid bulk modulus, crack porosity, and/or frequency is reduced to zero. It is shown that squirt flow is incapable of explaining the measured level of loss (10-2 < Q-1 < 10-1) within the seismic band of frequencies (1 to 104 Hz); however, either of the two mesoscopic scale models easily produce enough attenuation to explain the field data.
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