True-amplitude, angle-domain, common-image gathers from one-way wave-equation migrations

True-amplitude wave-equation migration provides a quality migrated image of the earth’s interior. In addition, the amplitude of the output provides an estimate of the angular-dependent reflection coefficient, similar to the output of Kirchhoff inversion. Recently, true-amplitude wave-equation migration for common-shot data has been proposed to generate amplitude-reliable, shot-domain, common-image gathers in heterogeneous media. We present a method to directly produce angle-domain common-image gathers from both common-shot and shot-receiver wave-equation migration. Generating true-amplitude, shot-domain, common-image gathers requires a deconvolution-type imaging condition using the ratio of the upgoing and downgoing wavefield, each downward-projected to the image point. Producing true-amplitude, angle-domain, common-image gathers requires, instead, the product of the upgoing wavefield and the complexconjugate of the downgoing wavefield in the imaging condition. Since multiplication is a more stable computational process than division, the new methods proposed provide more stable ways of inverting seismic data. Furthermore, the resulting common-image gathers can be directly used for migrated amplitude-variation-with angle analysis and tomography-based velocity analysis. Shot-receiver wave-equation migration requires new true-amplitude, one-way wave equations with one depth variable and transverse variables for the coordinates corresponding to sources and receivers, hence, two transverse coordinates in 2D and four transverse coordinates in 3D. We propose a modified double-square-root one-way wave equation to produce true amplitude common-image angle gathers. We also demonstrate the new methods with some synthetic examples. Some numerical examples show that the new methods we propose give better amplitude performance on the migrated angle gathers.

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Effective models for the multidimensional wave equation in heterogeneous media over long time and numerical homogenization

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202516500627 ◽

2016 ◽

Vol 26 (14) ◽

pp. 2651-2684 ◽

Cited By ~ 5

Author(s):

Assyr Abdulle ◽

Timothée Pouchon

Keyword(s):

Wave Equation ◽

Wave Equations ◽

Heterogeneous Media ◽

Multiscale Model ◽

Bloch Wave ◽

Time Interval ◽

Numerical Homogenization ◽

Effective Equation ◽

Long Time ◽

Effective Models

A family of effective equations that capture the long time dispersive effects of wave propagation in heterogeneous media in an arbitrary large periodic spatial domain [Formula: see text] is proposed and analyzed. For a wave equation with highly oscillatory periodic spatial tensors of characteristic length [Formula: see text], we prove that the solution of any member of our family of effective equations is [Formula: see text]-close to the true oscillatory wave over a time interval of length [Formula: see text] in a norm equivalent to the [Formula: see text] norm. We show that the previously derived effective equation in [T. Dohnal, A. Lamacz and B. Schweizer, Bloch-wave homogenization on large time scales and dispersive effective wave equations, Multiscale Model. Simulat. 12 (2014) 488–513] belongs to our family of effective equations. Moreover, while Bloch wave techniques were previously used, we show that asymptotic expansion techniques give an alternative way to derive such effective equations. An algorithm to compute the tensors involved in the dispersive equation and allowing for efficient numerical homogenization methods over long time is proposed.

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Case Study: Improving the quality of the seismic reflection image for a Fujian mineral exploration data set with offset-domain common-image gathers

Geophysics ◽

10.1190/geo2019-0770.1 ◽

2021 ◽

pp. 1-45

Author(s):

Guofeng Liu ◽

Xiaohong Meng ◽

Johanes Gedo Sea

Keyword(s):

Wave Equation ◽

Image Quality ◽

Seismic Data ◽

Seismic Reflection ◽

Mineral Exploration ◽

Velocity Model ◽

Image Point ◽

Single Shot ◽

Data Set ◽

Depth Migration

Seismic reflection is a proven and effective method commonly used during the exploration of deep mineral deposits in Fujian, China. In seismic data processing, rugged depth migration based on wave-equation migration can play a key role in handling surface fluctuations and complex underground structures. Because wave-equation migration in the shot domain cannot output offset-domain common-image gathers in a straightforward way, the use of traditional tools for updating the velocity model and improving image quality can be quite challenging. To overcome this problem, we employed the attribute migration method. This worked by sorting the migrated stack results for every single-shot gather into the offset gathers. The value of the offset that corresponded to each image point was obtained from the ratio of the original migration results to the offset-modulated shot-data migration results. A Gaussian function was proposed to map every image point to a certain range of offsets. This helped improve the signal-to-noise ratio, which was especially important in handing low quality seismic data obtained during mineral exploration. Residual velocity analysis was applied to these gathers to update the velocity model and improve image quality. The offset-domain common-image gathers were also used directly for real mineral exploration seismic data with rugged depth migration. After several iterations of migration and updating the velocity, the proposed procedure achieved an image quality better than the one obtained with the initial velocity model. The results can help with the interpretation of thrust faults and deep deposit exploration.

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True-amplitude Gaussian-beam migration

Geophysics ◽

10.1190/1.3052116 ◽

2009 ◽

Vol 74 (2) ◽

pp. S11-S23 ◽

Cited By ~ 125

Author(s):

Samuel H. Gray ◽

Norman Bleistein

Keyword(s):

Wave Equation ◽

Gaussian Beam ◽

Opening Angle ◽

Amplitude Wave ◽

Amplitude Variation With Offset ◽

Imaging Condition ◽

Depth Migration ◽

Kirchhoff Migration ◽

Beam Depth ◽

Gaussian Beam Migration

Gaussian-beam depth migration and related beam migration methods can image multiple arrivals, so they provide an accurate, flexible alternative to conventional single-arrival Kirchhoff migration. Also, they are not subject to the steep-dip limitations of many (so-called wave-equation) methods that use a one-way wave equation in depth to downward-continue wavefields. Previous presentations of Gaussian-beam migration have emphasized its kinematic imaging capabilities without addressing its amplitude fidelity. We offer two true-amplitude versions of Gaussian-beam migration. The first version combines aspects of the classic derivation of prestack Gaussian-beam migration with recent results on true-amplitude wave-equation migration, yields an expression involving a crosscorrelation imaging condition. To provide amplitude-versus-angle (AVA) information, true-amplitude wave-equation migration requires postmigration mapping from lateral distance (between image location and source location) to subsurface opening angle. However, Gaussian-beam migration does not require postmigration mapping to provide AVA data. Instead, the amplitudes and directions of the Gaussian beams provide information that the migration can use to produce AVA gathers as part of the migration process. The second version of true-amplitude Gaussian-beam migration is an expression involving a deconvolution imaging condition, yielding amplitude-variation-with-offset (AVO) information on migrated shot-domain common-image gathers.

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Wide-azimuth angle gathers for wave-equation migration

Geophysics ◽

10.1190/1.3560519 ◽

2011 ◽

Vol 76 (3) ◽

pp. S131-S141 ◽

Cited By ~ 33

Author(s):

Paul Sava ◽

Ioan Vlad

Keyword(s):

Time Lag ◽

Computational Cost ◽

Azimuth Angle ◽

Image Point ◽

Velocity Analysis ◽

Amplitude Variation ◽

Imaging Condition ◽

Migration Velocity Analysis ◽

Migration Method ◽

Complex Geology

Extended common-image-point (CIP) gathers contain all of the necessary information for decomposition of reflectivity as a function of the reflection and azimuth angles at selected locations in the subsurface. This decomposition operates after the imaging condition applied to wavefields reconstructed by any type of wide-azimuth migration method, e.g., using downward continuation or time reversal. The reflection and azimuth angles are derived from the extended images using analytic relations between the space-lag and time-lag extensions. The transformation amounts to a linear Radon transform applied to the CIPs obtained after applying the extended imaging condition. If information about the reflector dip is available at the CIP locations, then only two components of the space-lag vectors are required, thus reducing computational cost and increasing the affordability of the method. Applications of this method include the study of subsurface illumination in areas of complex geology where ray-based methods are not usable and the study of amplitude variation with reflection and azimuth angles if the subsurface illumination is sufficiently dense. Migration velocity analysis could also be implemented in the angle domain, although an equivalent implementation in the extended domain is less costly and more effective.

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Joint anisotropic amplitude variation with offset inversion of PP and PS seismic data

Geophysics ◽

10.1190/geo2016-0516.1 ◽

2018 ◽

Vol 83 (2) ◽

pp. N31-N50 ◽

Cited By ~ 8

Author(s):

Jun Lu ◽

Yun Wang ◽

Jingyi Chen ◽

Ying An

Keyword(s):

Seismic Data ◽

Local Minimum ◽

Synthetic Data ◽

Inversion Method ◽

Amplitude Variation ◽

Amplitude Variation With Offset ◽

Avo Inversion ◽

Time Migration ◽

Amplitude Variation With Angle ◽

Two Stages

With the increase in exploration target complexity, more parameters are required to describe subsurface properties, particularly for finely stratified reservoirs with vertical transverse isotropic (VTI) features. We have developed an anisotropic amplitude variation with offset (AVO) inversion method using joint PP and PS seismic data for VTI media. Dealing with local minimum solutions is critical when using anisotropic AVO inversion because more parameters are expected to be derived. To enhance the inversion results, we adopt a hierarchical inversion strategy to solve the local minimum solution problem in the Gauss-Newton method. We perform the isotropic and anisotropic AVO inversions in two stages; however, we only use the inversion results from the first stage to form search windows for constraining the inversion in the second stage. To improve the efficiency of our method, we built stop conditions using Euclidean distance similarities to control iteration of the anisotropic AVO inversion in noisy situations. In addition, we evaluate a time-aligned amplitude variation with angle gather generation approach for our anisotropic AVO inversion using anisotropic prestack time migration. We test the proposed method on synthetic data in ideal and noisy situations, and find that the anisotropic AVO inversion method yields reasonable inversion results. Moreover, we apply our method to field data to show that it can be used to successfully identify complex lithologic and fluid information regarding fine layers in reservoirs.

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Amplitude-variation-with-offset and prestack-waveform inversion: A direct comparison using a real data example from the Rock Springs Uplift, Wyoming, USA

Geophysics ◽

10.1190/geo2014-0233.1 ◽

2015 ◽

Vol 80 (2) ◽

pp. B45-B59 ◽

Cited By ~ 32

Author(s):

Subhashis Mallick ◽

Samar Adhikari

Keyword(s):

Wave Equation ◽

Seismic Data ◽

Quantitative Estimation ◽

Waveform Inversion ◽

Forward Modeling ◽

Amplitude Variation ◽

Data Set ◽

Amplitude Variation With Offset ◽

Rock Springs Uplift ◽

Rock Springs

Recent advances in seismic data acquisition and processing allow routine extraction of offset-/angle-dependent reflection amplitudes from prestack seismic data for quantifying subsurface lithologic and fluid properties. Amplitude-variation-with-offset (AVO) inversion is the most commonly used practice for such quantification. Although quite successful, AVO has a few shortcomings primarily due to the simplicity in its inherent assumptions, and for any quantitative estimation of reservoir properties, they are generally interpreted in combination with other information. In recent years, waveform-based inversions have gained popularity in reservoir characterization and depth imaging. Going beyond the simple assumptions of AVO and using wave equation solutions, these methods have been effective in accurately predicting the subsurface properties. Developments of these waveform inversions have so far been along two lines: (1) the methods that use a locally 1D model of the subsurface for each common midpoint and use an analytical solution to the wave equation for forward modeling and (2) the methods that do not make any 1D assumption but use an approximate numerical solution to the wave equation in 2D or 3D for forward modeling. Routine applications of these inversions are, however, still computationally demanding. We described a multilevel parallelization of elastic-waveform inversion methodology under a 1D assumption that allowed its application in a reasonable time frame. Applying AVO and waveform inversion on a single data set from the Rock Springs Uplift, Wyoming, USA, and comparing them with one another, we also determined that the waveform-based method was capable of obtaining a much superior description of subsurface properties compared with AVO. We concluded that the waveform inversions should be the method of choice for reservoir property estimation as high-performance computers become commonly available.

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Model parameterization and amplitude variation with angle and azimuthal inversion in orthotropic media

Geophysics ◽

10.1190/geo2018-0778.1 ◽

2020 ◽

Vol 86 (1) ◽

pp. R1-R14

Author(s):

Zhaoyun Zong ◽

Lixiang Ji

Keyword(s):

Parameter Estimation ◽

Seismic Data ◽

Inversion Method ◽

Initial Model ◽

Amplitude Variation ◽

Inversion Result ◽

Model Parameterization ◽

Orthotropic Media ◽

The Stability ◽

Amplitude Variation With Angle

Horizontal layered formations with a suite of vertical or near-vertical fractures are usually assumed to be an approximate orthotropic medium and are more suitable for estimating fracture properties with wide-azimuth prestack seismic data in shale reservoirs. However, the small contribution of anisotropic parameters to the reflection coefficients highly reduces the stability of anisotropic parameter estimation by using seismic inversion approaches. Therefore, a novel model parameterization approach for the reflectivity and a pragmatic inversion method are proposed to enhance the stability of the inversion for orthotropic media. Previous attempts to characterize orthotropic media properties required using four or ﬁve independent parameters. However, we have derived a novel formulation that reduces the number of parameters to three. The inversion process is better conditioned with fewer degrees of freedom. An accuracy comparison of our formula with the previous ones indicates that our approach is sufﬁciently precise for reasonable parameter estimation. Furthermore, a Bayesian inversion method is developed that uses the amplitude variation with angle and azimuth (AVAZ) of the seismic data. Smooth background constraints reduce the similarity between the inversion result and the initial model, thereby reducing the sensitivity of the initial model to the inversion result. Cauchy and Gaussian probability distributions are used as prior constraints on the model parameters and the likelihood function, respectively. These ensure that the results are within the range of plausibility. Synthetic examples demonstrate that the adopted orthotropic AVAZ inversion method is feasible for estimating the anisotropic parameters even with moderate noise. The ﬁeld data example illustrates the inversion robustness and stability of the adopted method in a fractured reservoir with a single well control.

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Asymptotic elastic Green’s tensor

Geophysics ◽

10.1190/1.1442536 ◽

1988 ◽

Vol 53 (7) ◽

pp. 992-994 ◽

Cited By ~ 3

Author(s):

Jack K. Cohen

Keyword(s):

Wave Equation ◽

Seismic Data ◽

Wave Equations ◽

Homogeneous Medium ◽

Acoustic Wave Equation ◽

Inversion Formulas ◽

Green's Tensor ◽

Two Component ◽

Elastic Wave Equations ◽

Green’S Tensor

The Green’s function for the acoustic wave equation has been an essential ingredient in obtaining frequency inversion formulas in the acoustic limit (Cohen and Bleistein, 1979; Clayton and Stolt, 1981; Beylkin, 1985; Cohen et al., 1986; Bleistein et al., 1987). Similarly, the Green’s tensor is required for inversions based on the elastic wave equations. Indeed, Kuo and Dai (1984) and Dai and Kuo (1986) have already used an approximation to this tensor to derive a migration result for two‐component seismic data in the case of a homogeneous medium.

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True amplitude wave equation migration arising from true amplitude one-way wave equations

Inverse Problems ◽

10.1088/0266-5611/19/5/307 ◽

2003 ◽

Vol 19 (5) ◽

pp. 1113-1138 ◽

Cited By ~ 94

Author(s):

Yu Zhang ◽

Guanquan Zhang ◽

Norman Bleistein

Keyword(s):

Wave Equation ◽

Wave Equations ◽

Amplitude Wave

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Validation of first-order diffraction theory for the traveltimes and amplitudes of propagating waves

Geophysics ◽

10.1190/1.2358412 ◽

2006 ◽

Vol 71 (6) ◽

pp. T167-T177 ◽

Cited By ~ 35

Author(s):

Jeroen Jocker ◽

Jesper Spetzler ◽

David Smeulders ◽

Jeannot Trampert

Keyword(s):

Seismic Data ◽

Heterogeneous Media ◽

Diffraction Theory ◽

Amplitude Variation ◽

First Order ◽

Background Medium ◽

Data Inversion ◽

Order Diffraction ◽

Propagating Waves ◽

Velocity Anomalies

Ultrasonic measurements of acoustic wavefields scattered by single spheres placed in a homogenous background medium (water) are presented. The dimensions of the spheres are comparable to the wavelength and the wavelength and represent both positive (rubber) and negative (teflon) velocity anomalies with respect to the background medium. The sensitivity of the recorded wavefield to scattering in terms of traveltime delay and amplitude variation is investigated. The results validate a linear (first-order) diffraction theory for wavefields propagating in heterogeneous media with anomaly contrasts on the order of [Formula: see text]. The diffraction theory is compared further with the exact results known from literature for scattering from an elastic sphere, formulated in terms of Legendre polynomials. To investigate the 2D case, the first-order scattering theory is tested against 2D elastic finite-difference calculations. As the presented theory involves a volume integral, it is applicable to any geometric shape, and the scattering object does not need to be spherical or any other specific symmetrical shape. Furthermore, it can be implemented easily in seismic data inversion schemes, which is illustrated with examples from seismic crosswell tomography and a reflection experiment.

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