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George A. McMechan

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Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG/AAPG/SEPM First International Meeting for Applied Geoscience & Energy, September 26–October 1, 2021

Paper Number: SEG-2021-3593600

Abstract

We propose a convolutional-neural-network-based gradientfree multi-parameter reflection full waveform inversion (multiparameter CNN-RFWI) to train convolutional neural networks (CNNs) from multi-parameter starting models and an RTM image, to invert for multi-parameter models. The multiparameter training models are iteratively recreated based on the prior features and the multi-parameter distributions in multi-parameter starting models (at the first iteration) or CNN-predicted multi-parameter models (at the following iterations). The multi-parameter CNN-RFWI iteratively shifts the distributions of the multi-parameter training models and the CNN-predicted multi-parameter models toward those of the unknown true multi-parameter models. The multiparameter CNN-RFWI has the following advantages: 1) no data- or image-domain misfit function is needed, so it reduces cycle-skipping and cross-talk issues in datafitting multi-parameter (reflection) FWI. 2) Higher-frequency reflection data are preferred (e.g., ≥20 Hz) to obtain a highresolution RTM image as CNN input data, so the deep part of the multi-parameter models can be inverted from a high-resolution reflection image and multi-parameter starting models. Synthetic tests on a portion of the Sigsbee2A P-wave velocity and density models show that multi-parameter CNNRFWI can invert for the P-wave velocity and density models more accurately than the corresponding starting models.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG/AAPG/SEPM First International Meeting for Applied Geoscience & Energy, September 26–October 1, 2021

Paper Number: SEG-2021-3594524

Abstract

Multi-mode dispersion curve picking is required by many approaches for shallow-subsurface characterization using surface waves, but is typically a labor-intensive process. Previously we automated the picking procedure using convolutional neural network (CNN) based machine learning (ML) method, which made industry-scale application possible in terms of efficiency. This study improves the picking algorithm in three ways: (1) it is more stable by using a new loss function which combines the conventional wavenumber differences and the wavenumber slope differences; (2) it is more efficient because we pretrain with synthetic data and then use transfer learning; (3) it is more complete because we estimate uncertainties in the picking using random dropout. These improvements make the picking algorithm more mature technically.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG/AAPG/SEPM First International Meeting for Applied Geoscience & Energy, September 26–October 1, 2021

Paper Number: SEG-2021-3581550

Abstract

One of the main bottlenecks in the application of deeplearning in exploration seismology is the lack of an unbiased training velocity model set with features similar to those in the unknown true velocity model. If the unknown true velocity model has different features from the training velocity model set, no matter how large is the training model set, it is always biased and causes severe overfitting in deep-learning LSRTM. We propose to reduce the bias by building training models based on prior features in the migration velocity model. A novel, spatially-constrained, hierarchical, k-means method captures these prior features. Therefore, the training velocity model set will share features similar to those of the unknown true model, if the migration velocity model has accurate lowwavenumbers. Then, we apply U-net convolutional neural network architecture to approximate the inverse Hessian in model-domain LSRTM (deep-learning LSRTM). Synthetic tests on a BP gas-cloud model shows that deep-learning LSRTM predicts the reflectivity model accurately from a lower-quality (sparse-shot) RTM image.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, October 11–16, 2020

Paper Number: SEG-2020-3427827

Abstract

Picking dispersion curves is required for many approaches for shallow-subsurface characterization using surface waves but is typically a labor-intensive process. Reliable, automatic picking would make these methods more efficient and practicable. We present a convolutional neural network (CNN) based machine learning (ML) approach to automatically pick the curves for the fundamental and higher modes along the two azimuths of any 2D seismic profile. The approach can be extended to 3D by applying 2D processing along multiple azimuths. Various attributes such as amplitudes, coherency, local phase velocity as well as frequency and wavenumber of dispersion curves are derived; different sub-sets of these are tested in the training process to assess the best combinations. We use the U-net architecture that is modified to convert the conventional 2D image segmentation problem in ( f , k ) domain into a direct multi-mode curve fitting and a subsequent picking process. The effectiveness of the automatic picking process is demonstrated in this study through applications to a field OBN dataset where different modes of Scholte waves were recorded. Presentation Date: Tuesday, October 13, 2020 Session Start Time: 1:50 PM Presentation Time: 4:45 PM Location: 351F Presentation Type: Oral

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, October 11–16, 2020

Paper Number: SEG-2020-3425921

Abstract

We implement elastic full wave-form inversion in the framework of recurrent neural networks. We use a staggered-grid stress-particle-velocity scheme to solve the elastodynamic equations for forward modeling. Automatic differentiation is obtained, with batch gradient descent optimization, for inversions of P- and S-wave velocities simultaneously. We analyze the inï¬‚uence of different batch sizes on the inversion, and ï¬nd that setting a minimum batch size (i.e., 1) has the best convergence rate for both clean and noisy data. The algorithm is tested with two synthetic models and show acceptable inversion results. Presentation Date: Wednesday, October 14, 2020 Session Start Time: 9:20 AM Presentation Time: 10:10 AM Location: Poster Station 3 Presentation Type: Poster

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, October 11–16, 2020

Paper Number: SEG-2020-3420598

Abstract

Gradient-descent FWI updates a velocity model by finding an optimal step length, which is then multiplied by the negative gradient to give a model perturbation to minimize the data residuals in a least-squares sense. We propose a CNN boosted FWI to apply a convolutional neural network (CNN) as a weighted weak approximator to represent the model perturbation to minimize the data residuals. In addition to finding the optimal step length, just as gradient-descent FWI does, CNN-boosted FWI fixes this optimal step length and optimizes the CNN, which is originally trained to approximate the negative gradients at each iteration, to update the velocity model. Synthetic examples using the modified Marmousi2 P wave model show that CNN-boosted FWI, as well as a hybrid, of CNN-boosted FWI and gradient-descent FWI, inverts for the velocity model with lower model and data errors than the gradient-descent FWI does. Presentation Date: Tuesday, October 13, 2020 Session Start Time: 1:50 PM Presentation Time: 2:40 PM Location: 351F Presentation Type: Oral

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, October 11–16, 2020

Paper Number: SEG-2020-3411534

Abstract

Strong surface and guided waves in land seismic data often distort signals from deeper structures, leading to poor S/N. Accurate modeling and attenuation of aliased surface waves has been a long-standing challenge in onshore seismic processing and imaging. To address this issue, we propose a method based on elastic full waveform inversion (FWI) using the spectral element seismic simulator to model and attenuate surface waves from the recorded data. This method first reconstructs surface waves by estimating a relatively shallow earth model via elastic multi-parameter FWI. The reconstructed surface waves are then used through an adaptive subtraction process to remove the surface waves from the recorded seismograms. The proposed method does not impose any 1-D assumption as conventional dispersionbased modeling approach does, and hence can model complicated wave phenomena, including back scattered surface waves. We demonstrate the performance of the proposed method using the SEAM II Arid model. Presentation Date: Tuesday, October 13, 2020 Session Start Time: 1:50 PM Presentation Time: 3:05 PM Location: Poster Station 13 Presentation Type: Poster

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, September 15–20, 2019

Paper Number: SEG-2019-3214913

Abstract

ABSTRACT Elastic reverse time migration (E-RTM) using full wave equation is able to generate both P-P and converted P-S images of the subsurface. Besides the ability of imaging complicated geological structures, similar to acoustic imaging, image artifacts are also observed in the P-P and P-S images, where high velocity contrasts/gradients in the velocity model produce reflections of the source and receiver wavefields. It has been proven that separating RTM images into different components based on propagation directions of the source and receiver wavefields is able to effectively remove such image artifacts. We first compare different image separation methods, including up/down image separation during, and post, imaging. The post-imaging method is simple, as it requires only simple operations applied to the stacked RTM images. The separated up/down images are similar to those separated during imaging, but there are also difference. The difference comes from different physical meanings of the up/down separated images. Furthermore, we present a new workflow that combines a cost efficient up/down imaging condition and a post-imaging separation method. A numerical example using the Sigsbee 2A model indicates that the up/down separated P-P and P-S images from the new workflow are equivalent to those by during imaging separation methods. Presentation Date: Wednesday, September 18, 2019 Session Start Time: 8:30 AM Presentation Start Time: 9:45 AM Location: 214C Presentation Type: Oral

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, September 15–20, 2019

Paper Number: SEG-2019-3215391

Abstract

ABSTRACT Conventional full waveform inversion (FWI) updates a velocity model, by minimizing the data residuals between the forward-propagating source wavefield and the forward time observed data, at the receiver positions. We propose an alternative inversion algorithm to update the velocity model by minimizing the virtual source artifacts in the source domain (SFWI). We first define a source-receiver wavefield, which is extrapolated by reinserting the forward-time observed data as boundary-value conditions, at the receiver positions, along with the forward-time propagation of the source wavefield from the source point, so the data residuals are forced to be zero at each time step. If the velocity model is incorrect, the source-receiver wavefield will contain artifact waves, which are created by the mismatch between the reinserted observed data and the extrapolated source wavefield. By extracting and minimizing virtual source artifacts, which is equivalent to creating the artifact waves, the velocity model will be iteratively updated toward the true velocity model. Tests on synthetic data show that the source-domain FWI inverts for the velocity model as accurately as the conventional FWI. Presentation Date: Monday, September 16, 2019 Session Start Time: 1:50 PM Presentation Time: 3:05 PM Location: Poster Station 9 Presentation Type: Poster

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the SEG International Exposition and Annual Meeting, September 15–20, 2019

Paper Number: SEG-2019-3216232

Abstract

ABSTRACT We develop and illustrate concurrent elastic inversion of Rayleigh and P and S body waves using interleaved envelope and waveform-based misfit functions. Rayleigh and body waves are both solutions of the same wave equation, but the physics of their interaction with the elastic parameters Vp, Vs and density are independent of each other and thus they provide independent constraints in the inversion. The FWI interleaves envelope-based and waveform-based misfit functions in a gradually-increasing frequency, multi-scale, inversion strategy. A wavelet and its envelope have different effective bandwidths that partially overlap in both frequency and wavenumber so the half-wavelength continuity criterion is satisfied, and is exploited to reduce cycle skipping and to increase resolution. Because the constraints are more decoupled in concurrent inversion of both Rayleigh and body waves, the ability to recover density is better than that obtained by other versions of FWI. Presentation Date: Wednesday, September 18, 2019 Session Start Time: 8:30 AM Presentation Time: 11:25 AM Location: 302B Presentation Type: Oral

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2018 SEG International Exposition and Annual Meeting, October 14–19, 2018

Paper Number: SEG-2018-2984679

Abstract

ABSTRACT Seismic full waveform modeling is computationally expensive for viscoelastic media. Seismic modeling for viscocoustic media is much cheaper, at the cost of incomplete physics. We present a method for modeling the viscoelastic effects of P waves, by using a modified viscoacoustic wave simulation. We derive a residual error source term when an viscoacoustic solution is used for approximating the solution of the viscoelastic equation, by comparing the viscoacoustic and viscoelastic wave equations. The residual error source term is used as the source for a second viscoacoustic simulation. The correctedparticle velocities of the P wave can be obtained by adding the wavefield from the second simulation to the original viscoacoustic wavefield. Only P waves are modeled. The overall cost is about twice that of viscoacoustic modeling, but is significantly less than a viscoelastic propagation, because thereare fewer equations, and we can use a coarser grid and larger time steps. Numerical examples show that the amplitudes of the P wave matches those from viscoelastic wave modeling. Presentation Date: Monday, October 15, 2018 Start Time: 1:50:00 PM Location: 205A (Anaheim Convention Center) Presentation Type: Oral

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2018 SEG International Exposition and Annual Meeting, October 14–19, 2018

Paper Number: SEG-2018-2998123

Abstract

The Poynting vector (PV) has been widely used to calculate propagation vectors of a pressure field (PF) in acoustic media. The most widely-used acoustic PV formula is the negative of a product of time and space derivatives. These two derivatives result in a phase-shift between the PF and its PV; particularly, for a PF at a local magnitude peak, its PV modulus is zero and thus the propagation direction there is undefined. This "zero-modulus" issue is not consistent with the physical definition of the PV, which is the directional energy flux density of a PF, because this definition indicates that the variation of the PV modulus should be consistent with the pressure magnitude. We derive the dynamically-correct PV formula for acoustic media, which is the negative of the product of the reciprocal of the density, the PF itself, and a factor that is obtained by applying both a time integration and a space derivative to the PF. This dynamically-correct PV does not suffer from the "zero-modulus" problem and we also use it to update the multidirectional PV (MPV), which produces a dynamically-correct MPV. Presentation Date: Thursday, October 18, 2018 Start Time: 8:30:00 AM Location: 207A (Anaheim Convention Center) Presentation Type: Oral

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2018 SEG International Exposition and Annual Meeting, October 14–19, 2018

Paper Number: SEG-2018-2998139

Abstract

ABSTRACT PP and PS images are important output from elastic reverse time migration. The calculation requires P and S wave mode separation, which can be generally divided into two categories, divergence & curl (D&C) operations and the decoupled extrapolation system. For the D&C operations, the advantage is that it naturally produces a scalar P particle velocity to give the PP image; the main difficulty is the polarity correction for the PS and SP images at normal incidence. For the decoupled extrapolation system, it produces vector wavefields and thus a vector-based imaging condition is required to produce scalar images. The advantage is that it naturally corrects the polarity reversal of the PS and SP images at normal incidence; the disadvantage is that the PP and PS images require calculation of propagation angles to compensate the polarity reversal at open angle 90°. By combining the scalar and vector imaging conditions, we propose an affordable scheme: the PP image is calculated by the scalar imaging condition; the PS image calculated by the vector imaging condition. This imaging strategy requires both the scalar and vector P particle velocities, and the vector S particle velocity. To obtain them, we find a relationship between the amplitude-and-phase-compensated D&C system (that produces a scalar P particle velocity) and the decoupled extrapolation system (that outputs vector P and S particle velocities), and then we use a single decoupled system to produce both the scalar and vector P particle velocities, and the vector S particle velocity. Thus, we are able to combine the scalar and vector imaging conditions to calculate PP and PS images efficiently. Presentation Date: Monday, October 15, 2018 Start Time: 1:50:00 PM Location: Poster Station 21 Presentation Type: Poster

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2018 SEG International Exposition and Annual Meeting, October 14–19, 2018

Paper Number: SEG-2018-2963265

Abstract

ABSTRACT Full waveform inversion (FWI) is useful to invert for seismic models (e.g., velocity and density) by matching simulated seismic data with recorded field data. Conventional FWI assumes that the true geologic model consists of a background smooth model (the starting model) and a perturbation model which generates the first-order scattering wave only (a firstorder Born approximation). This assumption is not valid for a geologic model containing a large geometrical target body with strong impedance contrast (e.g., salt body), since most of the downgoing incident energy will be reflected at its top. Therefore, the data misfit is more sensitive to the waves reflected at the top of the geometrical target than the waves generated inside it. As a result, compared with the inner part and the bottom boundary of the geometrical target, the top boundary is more likely to be inverted accurately. In this paper, we modify FWI by implementing a convolutional neural network (CNN-FWI) to capture and then invert the geometrical target body. Synthetic tests on the Sigsbee P-wave model shows that the proposed CNN-FWI can invert for the salt body geometry more accurately than conventional gradient-descent FWI (GD-FWI) does. Presentation Date: Wednesday, October 17, 2018 Start Time: 1:50:00 PM Location: 204B (Anaheim Convention Center) Presentation Type: Oral

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2015 SEG Annual Meeting, October 18–23, 2015

Paper Number: SEG-2015-5824203

Abstract

Summary Without considering intrinsic attenuation, reverse time migration (RTM) in lossy media may produce unfocused migration images, because of the amplitude loss and velocity dispersion in the recorded data caused by Q. We use a constant Q viscoelastic equation for source and receiver extrapolations. Two fractional Laplacian operators introduce velocity dispersion and amplitude loss effects, respectively, for each wave mode. Velocity dispersion and amplitude loss can be separated. To compensate the Q effect, we reverse the sign of the amplitude loss operator, and keep the sign of velocity dispersion operator unchanged, during receiver extrapolations. Boundary-valuebased viscoelastic source wavefield reconstruction shows that this Q-compensation strategy can compensate the amplitude loss and avoid phase distortion. The source-normalized crosscorrelation imaging condition is applied. Two numerical examples show that, with Q compensation, structures are better focused at the correct positions, and with more energy. Introduction Multi-parameter imaging in viscous media is of growing interest (Prieux, et al., 2013). For viscoelastic media, energy absorption and intrinsic velocity dispersion affects the amplitude and travel time of the wavefield. For seismic imaging in viscous media, without considering Q, migrated images will not be well focused (Zhang et al., 2010; Zhu et al., 2014), because of the amplitude loss along the propagation path, and from the phase distortion. One approach to compensate the Q effects is in the data domain. A simple and straightforward method is by applying an inverse Q filter to the seismograms (Hargreaves and Calvert, 1991). Inverse Q filtering is based on a 1-D assumption; it is not applicable to Q compensation in complicated media. Since seismic attenuation affects waveforms during propagation, it is more natural and accurate to compensate the Q effects during wavefield extrapolation, by modifying the wavefield propagators. To include Q effects in source and receiver wavefield extrapolations, two categories of viscous wave equations are widely used. The first is based on the generalized standard linear solid model (e.g. Carcione et al., 1988). Memory variables are introduced to address the computation of the convolutional kernel in the stress-strain constitutive relations, and thus introduce the Q effects. For viscoelastic RTM, Deng and McMechan (2008) proposed to reverse the sign of memory variables, during receiver extrapolation, to compensate the amplitude loss.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2015 SEG Annual Meeting, October 18–23, 2015

Paper Number: SEG-2015-5838306

Abstract

Summary Angle gathers obtained from reverse-time migration are important for true-amplitude migration, migration velocity analysis, and angle-dependent inversion. Among the existing methods to compute angle gathers, those using the direction vectors in the time-space (t-x) domain are efficient, such as the Poynting vector, which gives the group velocity direction. However, the Poynting vector can give only one direction per grid point per time step and cannot compute multiple directions when wavefields overlap. We use the t-x slowness vector to compute the phase velocity direction, and find an important relationship between the slowness vector in t-x and the plane-wave decomposition in the frequency-wavenumber (?-k) domain: they can be transformed into each other by Fourier transforms (FTs) between t-x and ?-k; the FT from t-x to ?-k decomposes the t-x slowness vector into different ? and k, and the inverse FT from ?-k to t-x sums all planewave components at different ? and k. To make the t-x slowness vector have multiple directions, we divide the inverse FT, from ?-k to t-x, into several parts by using k and the sign of ?, and compute one slowness vector for each of the separated wavefields; the set of these singledirection slowness vectors is a multi-direction slowness vector. Introduction Reverse-time migration (RTM) has achieved considerable progress in the last decade. An important output from RTM is the angle-domain common-image gathers (ADCIGs). Among the methods to compute angle gathers, using the Poynting vector (Cervený, 2001; Yoon et al., 2004) is efficient. However, the Poynting vector is not stable, mainly because it can give only one direction per grid point per time step. Up to now, there is still not a solution. We suggest to use the slowness vector in t-x to take place of the Poynting vector, and prove that their expressions are same for an acoustic wavefront in an isotropic medium. We develop the multi-direction slowness vector to compute the correct multiple directions for the overlapping wavefields.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2015 SEG Annual Meeting, October 18–23, 2015

Paper Number: SEG-2015-5838618

Abstract

Summary Multi-direction slowness vectors can compute multiple directions when wavefields overlap. It requires full wave decomposition by using the wavenumber and the sign of the angular frequency, and then computes a slowness vector component for each separated wavefield. In this method, the decomposition in the frequency-wavenumber (&#969:-k) domain needs to be done for both the positive- and negative-frequency wavefields, and the Fourier transforms between the time and frequency domains require large I/O time. To solve these problems, we prove that the complexvalued decomposition results by using either positive- or negative-frequency wavefields are conjugate, so we need only the positive-frequency wavefields, which can be obtained by the complex-valued extrapolation of the positive-frequency source wavelet and recorded data in the time-space (t-x) domain; therefore, we do the approximate wave decomposition in t-k instead of &#969:-k. The complexvalued extrapolation is twice the computational complexity as the real-valued extrapolation. Besides, because the slowness vector component for each separated wavefield is a product of the time and space derivatives, the direction of the wavefield at a peak or a trough is unstable. We use a variable time-shift to solve this problem. Introduction The multi-direction slowness vector (Tang and McMechan, 2015) can give multiple directions when the wavefields overlap. The main cost in this method is the wave decomposition that uses the k and the sign of &#969:, which can be done in &#969:-k (Hu and McMechan, 1987). Because we need only the sign of &#969: and there are only two possibilities (positive or negative), it is better to separate the wavefields into two complex-valued parts with either positive or negative frequencies first, and then decompose them in t-k. There are two problems: (I) The Fourier transforms (FTs) between time and frequency domains is a great I/O challenge. The wavefields are simulated in time, so unless we store the wavefields at all the time steps in the memory, we need a huge I/O time to reorder the wavefields for the FTs between time and frequency domains; (II) We need to do the wave decomposition in the wavenumber domain for both of the two complex-valued parts with either positive or negative frequencies.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2014 SEG Annual Meeting, October 26–31, 2014

Paper Number: SEG-2014-0070

Abstract

Summary Anisotropic attenuation is important, especially in fluidsaturated reservoirs with high fracture density. Wave induced meso-scale fluid flow is considered to be the major cause of intrinsic attenuation at seismic frequencies. We perform a sensitivity test of anisotropic attenuation and velocity, to reservoir parameters in fractured HTI media based on the meso-scale fluid flow mechanism. The visco-elastic T-matrix, a unified effective medium theory of global and local fluid flow mechanisms, is used to compute frequency-dependent anisotropic attenuation and velocity for ranges of reservoir properties, including fracture density, inclusion aspect ratio, fluid type and permeability; anisotropic attenuation is sensitive to these properties and thus is a potential tool for reservoir characterization.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2012 SEG Annual Meeting, November 4–9, 2012

Paper Number: SEG-2012-0402

Abstract

SUMMARY A new algorithm for wave-equation migration velocity analysis is proposed. Wavefield extrapolation is done with a two-way wave-equation in reverse-time migration, and the angles in the angle-domain common gathers (ADCIGs) are extracted by a polarization method without ray tracing. In the ADCIGs, the residual moveout in depth is used to build a linear system for velocity updating. Wavepaths (which are similar to the raypaths in ray tomography) are constructed from the source-only wavefield polarizations (without ray tracing). In the linear equations, the wavepaths in depth and the residual moveouts, are connected at the reflection point; by solving a linear system, the velocity model can be updated. The convergence condition is flattening of the ADCIGs. A layered synthetic model with a fault successfully illustrates the procedure.

Proceedings Papers

Publisher: Society of Exploration Geophysicists

Paper presented at the 2011 SEG Annual Meeting, September 18–23, 2011

Paper Number: SEG-2011-3130

Abstract

ABSTRACT We propose an alternative (new) method to produce common image gathers in the incident-angle domain by calculating wavenumbers directly from the P-wave polarization rather than using the dominant wavenumber, for obtaining the normal to the source wavefront. In isotropic acoustic media, the wave propagation direction can be directly calculated as the spatial gradient direction of the acoustic wavefield, which is parallel to the wavenumber direction (the normal to the wavefront). Instantaneous wavenumber, obtained via a novel Hilbert transform approach, is used to calculate the local normal to the reflectors in a stacked migrated image. The local incident angle is produced as the difference between the propagation direction and the normal to the reflector. By reordering the migrated images (over all common-source gathers) by incident angle, common-image gathers are produced in the incident-angle domain. P- and S-wave separations allow both PP and PS common-image gathers to be calculated in the angle domain. Unlike the space-shift image condition for calculating the common-image gather in the angle domain, we use a cross-correlation image condition, which is substantially more efficient. The proposed method is a direct method, and the resolution of the local incident angles is higher than those of Fourier transform-based methods. In addition, this method is less dependent on the data quality than the space-shift method. The concepts are successfully implemented and illustrated with 2D synthetic acoustic and elastic examples.

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