Amplitude effect of the free surface in elastic reverse-time extrapolation

Geophysics ◽  
2008 ◽  
Vol 73 (5) ◽  
pp. S177-S184 ◽  
Author(s):  
Robert Sun ◽  
George A. McMechan
Keyword(s):  
Free Surface ◽  
Computational Model ◽  
P Wave ◽  
Surface Boundary ◽  
Reverse Time ◽  
S Wave ◽  
Elastic Data ◽  
Converted Waves ◽  
Incident Waves ◽  
Time Extrapolation

We evaluate the physical validity of surface boundary conditions of the computational model in reverse-time extrapolation of 3D, three-component (3-C) elastic seismic data acquired at the earth’s free surface by using mathematical derivations and numerical simulations. Reverse-time extrapolation of elastic data assumes that only the incident P- or S-waves are reconstructed during extrapolation into the computational grid. However, superposition of the (upgoing) incident waves and the (downgoing) reflected and converted waves generated at the free surface also is recorded in data acquisition and is input into reverse-time extrapolation. In elastic reverse-time extrapolation, the computational model needs to have an absorbing top boundary. When the 3D, 3-C elastic data are inserted into the computational model during reverse-time extrapolation, the originally incident P- or S-wave is reconstructed. In addition, the free-surface P-to-P reflected and P-to-S converted waves recombine to reconstruct a second incident P-wave, and the free-surface S-to-S reflected and S-to-P converted waves recombine to reconstruct a second incident S-wave. Therefore, 3D elastic reverse-time extrapolation reconstructs the incident waves with displacement amplitudes increased by a fixed factor of exactly two when free-surface reflections and conversions are in the data. In this implementation, reconstructed (virtual) waves propagating upward from the free surface enter an absorbing zone and disappear.

Scalar reverse‐time depth migration of prestack elastic seismic data

Geophysics ◽  
2001 ◽  
Vol 66 (5) ◽  
pp. 1519-1527 ◽  
Author(s):  
Robert Sun ◽  
George A. McMechan
Keyword(s):  
Seismic Waves ◽  
Velocity Model ◽  
P Wave ◽  
Common Source ◽  
Reverse Time ◽  
S Wave ◽  
Wave Source ◽  
Reflected Waves ◽  
S Waves ◽  
Elastic Data

Reflected P‐to‐P and P‐to‐S converted seismic waves in a two‐component elastic common‐source gather generated with a P‐wave source in a two‐dimensional model can be imaged by two independent scalar reverse‐time depth migrations. The inputs to migration are pure P‐ and S‐waves that are extracted by divergence and curl calculations during (shallow) extrapolation of the elastic data recorded at the earth’s surface. For both P‐to‐P and P‐to‐S converted reflected waves, the imaging time at each point is the P‐wave traveltime from the source to that point. The extracted P‐wave is reverse‐time extrapolated and imaged with a P‐velocity model, using a finite difference solution of the scalar wave equation. The extracted S‐wave is reverse‐time extrapolated and imaged similarly, but with an S‐velocity model. Converted S‐wave data requires a polarity correction prior to migration to ensure constructive interference between data from adjacent sources. Synthetic examples show that the algorithm gives satisfactory results for laterally inhomogeneous models.

Prestack reverse time imaging in tunnels based on the decoupled nonconversion elastic-wave equation

Geophysics ◽  
2020 ◽  
Vol 85 (6) ◽  
pp. S371-S383
Author(s):  
Yuxiao Ren ◽  
Zhichao Yang ◽  
Bin Liu ◽  
Xinji Xu ◽  
Yangkang Chen
Keyword(s):  
Seismic Data ◽  
Accurate Method ◽  
Wave Mode ◽  
P Wave ◽  
Tunnel Construction ◽  
Reverse Time ◽  
S Wave ◽  
Tunnel Face ◽  
Time Migration ◽  
Converted Waves

The safety and efficiency of tunnel construction depend on the knowledge of complex geologic conditions. Hence, an accurate forward-prospecting technique is required to detect unexpected inhomogeneous geologic structures ahead of tunnel construction. As an accurate method to image geologic heterogeneities, elastic reverse time migration (ERTM) is introduced to the field of tunnel forward prospecting. However, in the tunnel environment, ERTM images may suffer from the interference of crosstalk artifacts, which are caused by converted waves on tunnel surfaces. Therefore, considering the actual influence of the tunnel body, the decoupled nonconversion elastic equation was incorporated into traditional ERTM. This method prevents the generation of converted waves but ensures independent P- and S-wave propagation. In addition, wave-mode separation for raw seismic data is required in our approach. Synthetic examples based on the real geologic environment of tunnels show that our method produces satisfactory results for P-wave and S-wave imaging and the S-wave can produce a better imaging effect in tunnels. Finally, we apply our method to the seismic data obtained from a real highway tunnel construction site to demonstrate its performance in real-world applications. The results indicate that the migrated images can help to accurately constrain the geologic formations ahead of the tunnel face.

An exact anelastic model for the free-surface relfection of P and S-I waves

1989 ◽  
Vol 79 (3) ◽  
pp. 842-859
Author(s):  
R. D. Borcherdt ◽  
G. Glassmoyer
Keyword(s):  
Free Surface ◽  
Volumetric Strain ◽  
Major Axis ◽  
P Wave ◽  
Reflection Coefficients ◽  
S Wave ◽  
Low Loss ◽  
Surface Reflection ◽  
Pierre Shale ◽  
Homogeneous Wave

Abstract Exact anelastic solutions incorporating inhomogeneous waves are used to model numerically S-I and P waves incident on the free surface of a low-loss anelastic half-space. Anelastic free-surface reflection coefficients are computed for the volumetric strain and displacement components of inhomogeneous wave fields. For the problem of an incident homogeneous S-I wave in Pierre shale, the largest strain and displacement amplitudes for the reflected P wave occur at angles of incidence for which the particle motion for the reflected inhomogeneous P wave is elliptical (minor/major axis = 0.6), the specific absorption (QP−1) is greater (300 per cent) and the velocity is less (25 per cent) than those for a corresponding homogeneous P wave, the direction of phase propagation is not parallel to the free surface, and the amplitude of the wave shows a significant increase with depth (6 per cent in one wavelength). Energy reflection coefficients computed for this low-loss anelastic model show that energy flow due to interaction of the incident and reflected waves reach maxima (30 per cent of the incident energy) near large but nongrazing angles of incidence. For the problem of an incident homogeneous P wave in Pierre shale, the inhomogeneity of the reflected S wave is shown not to contribute to significant variations in wave field characteristics over those that would be expected for a homogeneous wave.

3-D elastic prestack, reverse‐time depth migration

Geophysics ◽  
1994 ◽  
Vol 59 (4) ◽  
pp. 597-609 ◽  
Author(s):  
Wen‐Fong Chang ◽  
George A. McMechan
Keyword(s):  
Finite Differences ◽  
Synthetic Data ◽  
P Wave ◽  
Common Source ◽  
Reverse Time ◽  
Depth Migration ◽  
Full Wave ◽  
Source Data ◽  
Recorded Data ◽  
Time Extrapolation

By combining and extending previous algorithms for 2-D prestack elastic migration and 3-D prestack acoustic migration, a full 3-D elastic prestack depth migration algorithm is developed. Reverse‐time extrapolation of the recorded data is by 3-D elastic finite differences; computation of the image time for each point in the 3-D volume is by 3-D acoustic finite differences. The algorithm operates on three‐component, vector‐wavefield common‐source data and produces three‐component vector reflectivity distributions. Converted P‐to‐S reflections are automatically imaged with the primary P‐wave reflections. There are no dip restrictions as the full wave equation is used. The algorithm is illustrated by application to synthetic data from three models; a flat reflector, a dipping truncated wedge overlying a flat reflector, and the classical French double dome and fault model.

Direct vector-field method to obtain angle-domain common-image gathers from isotropic acoustic and elastic reverse time migration

Geophysics ◽  
2011 ◽  
Vol 76 (5) ◽  
pp. WB135-WB149 ◽  
Author(s):  
Qunshan Zhang ◽  
George A. McMechan
Keyword(s):  
Direct Method ◽  
Incident Angle ◽  
Field Method ◽  
Propagation Direction ◽  
P Wave ◽  
Common Source ◽  
Spatial Gradient ◽  
Reverse Time ◽  
S Wave ◽  
Time Migration

We have developed 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 as 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 the 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) with incident angle, common-image gathers are produced in the incident-angle domain. Instantaneous wavenumber takes the place of the normal to the reflector in the migrated image. 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 angle domain, we use the crosscorrelation image condition, which is substantially more efficient. This is a direct method, and is less dependent on the data quality than the space-shift method. The concepts were successfully implemented and tested with 2D synthetic acoustic and elastic examples, including a complicated (Marmousi2) model that illustrates effects of multipathing in angle-domain common-image gathers.

Acoustic wave propagation in tilted transversely isotropic media: Incorporating topography

Geophysics ◽  
2016 ◽  
Vol 81 (5) ◽  
pp. C265-C278 ◽  
Author(s):  
Jeffrey Shragge
Keyword(s):  
Wave Propagation ◽  
Free Surface ◽  
Surface Topography ◽  
Domain Boundary ◽  
Waveform Inversion ◽  
Transversely Isotropic ◽  
Surface Boundary ◽  
Reverse Time ◽  
Time Migration ◽  
Tti Media

Simulating two-way acoustic wavefield propagation directly from a free-surface boundary in the presence of topography remains a computational challenge for applications of reverse time migration (RTM) or full-waveform inversion (FWI). For land-seismic settings involving heavily reworked geology (e.g., fold and thrust belts), two-way wavefield propagation operators should also handle commonly observed complex anisotropy including tilted transversely isotropic (TTI) media. To address these issues, I have extended a system of coupled partial differential equations used to model 3D acoustic TTI wave propagation in Cartesian coordinates to more generalized 3D geometries, including a deformed computational mesh with a domain boundary conformal to free-surface topography. A generalized curvilinear transformation is used to specify a system of equations governing 3D acoustic TTI wave propagation in the “topographic” coordinate system. The developed finite-difference time-domain numerical solution adapts existing Cartesian TTI operators to this more generalized geometry with little additional computational overhead. Numerical evaluations illustrate that 2D and 3D impulse responses are well-matched to those simulated on Cartesian meshes and analytic traveltimes for homogeneous elliptical TTI media. Accordingly, these generalized acoustic TTI propagators and their numerical adjoints are useful for undertaking most RTM or FWI applications using computational domains conforming to free-surface topography.

Tensorial elastodynamics for isotropic media

Geophysics ◽  
2020 ◽  
Vol 85 (6) ◽  
pp. T359-T373
Author(s):  
Jeffrey Shragge ◽  
Tugrul Konuk
Keyword(s):  
Free Surface ◽  
Surface Topography ◽  
Numerical Solutions ◽  
Waveform Inversion ◽  
Real Life ◽  
Staggered Grid ◽  
Surface Boundary ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Time Migration

Numerical solutions of 3D isotropic elastodynamics form the key computational kernel for many isotropic elastic reverse time migration and full-waveform inversion applications. However, real-life scenarios often require computing solutions for computational domains characterized by non-Cartesian geometry (e.g., free-surface topography). One solution strategy is to compute the elastodynamic response on vertically deformed meshes designed to incorporate irregular topology. Using a tensorial formulation, we have developed and validated a novel system of semianalytic equations governing 3D elastodynamics in a stress-velocity formulation for a family of vertically deformed meshes defined by Bézier interpolation functions between two (or more) nonintersecting surfaces. The analytic coordinate definition also leads to a corresponding analytic free-surface boundary condition (FSBC) as well as expressions for wavefield injection and extraction. Theoretical examples illustrate the utility of the tensorial approach in generating analytic equations of 3D elastodynamics and the corresponding FSBCs for scenarios involving free-surface topography. Numerical examples developed using a fully staggered grid with a mimetic finite-difference formulation demonstrate the ability to model the expected full-wavefield behavior, including complex free-surface interactions.

Elastic inversion of near- and postcritical reflections using phase variation with angle

Geophysics ◽  
2012 ◽  
Vol 77 (4) ◽  
pp. R149-R159 ◽  
Author(s):  
Xinfa Zhu ◽  
George A. McMechan
Keyword(s):  
Wave Velocity ◽  
Spherical Wave ◽  
Phase Variation ◽  
P Wave ◽  
Wide Angle ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Time Migration ◽  
Elastic Inversion

Near- and postcritical (wide-angle) reflections provide the potential for velocity and density inversion because of their large amplitudes and phase-shifted waveforms. We tested using phase variation with angle (PVA) data in addition to, or instead of, amplitude variation with angle (AVA) data for elastic inversion. Accurate PVA test data were generated using the reflectivity method. Two other forward modeling methods were also investigated, including plane-wave and spherical-wave reflection coefficients. For a two half-space model, linearized least squares was used to invert PVA and AVA data for the P-wave velocity, S-wave velocity, and the density of the lower space and the S-wave velocity of the upper space. Inversion tests showed the feasibility and robustness of PVA inversion. A reverse-time migration test demonstrated better preservation of PVA information than AVA information during wavefield propagation through a layered overburden. Phases of deeper reflections were less affected than amplitudes by the transmission losses, which makes the results of PVA inversion more accurate than AVA inversion in multilayered media. PVA brings useful information to the elastic inversion of wide-angle reflections.

Amplitude-preserving scalar PP and PS imaging condition for elastic reverse time migration based on a wavefield decoupling method

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. S113-S125 ◽  
Author(s):  
Xiyan Zhou ◽  
Xu Chang ◽  
Yibo Wang ◽  
Zhenxing Yao
Keyword(s):  
Wave Vector ◽  
Numerical Models ◽  
P Wave ◽  
Imaging Method ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Decoupling Method ◽  
Time Migration ◽  
Imaging Results

To eliminate crosstalk within the imaging results of elastic reverse time migration (ERTM), we can separate the coupled P- and S-waves from the forward source wavefield and the backpropagated receiver wavefield. The P- and S-wave decoupling method retains the original phase, amplitude, and physical meaning in the separated wavefields. Thus, it is a vital wavefield separation method in ERTM. However, because these decomposed wavefields are vectors, we could consider how to retrieve scalar images that reveal the real reflectivity of the subsurface. For this purpose, we derive a scalar P-wave equation from the velocity-stress relationship for PP imaging. The phase and amplitude of this scalar P-wave are consistent with the scalarized P-wave. Therefore, this scalar P-wave can be exploited to perform PP imaging directly, with the imaging result retaining the amplitude characteristics. For PS imaging, it is difficult to calculate a dynamic preserved scalar S-wave. However, we have developed a scalar PS imaging method that divides the PS image into energy and sign components according to the geometric relationship between the wavefield vibration and propagation directions. The energy is calculated through the amplitude crosscorrelation of the forward P-wave and backpropagated S-wave from the receivers. The sign is obtained from the dot product of the forward P-wave vector and the backpropagated S-wave vector. These PP and PS imaging methods are suitable for 2D and 3D isotropic media and maintain the correct amplitude information while eliminating polarity-reversal phenomena. Several numerical models are used to verify the robustness and effectiveness of our method.

Elastic reflection waveform inversion based on the decomposition of sensitivity kernels

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. R235-R250 ◽  
Author(s):  
Zhiming Ren ◽  
Zhenchun Li ◽  
Bingluo Gu
Keyword(s):  
Wave Velocity ◽  
Waveform Inversion ◽  
Velocity Model ◽  
P Wave ◽  
Background Model ◽  
Reverse Time ◽  
S Wave ◽  
Velocity Models ◽  
Time Migration ◽  
Starting Model

Full-waveform inversion (FWI) has the potential to obtain an accurate velocity model. Nevertheless, it depends strongly on the low-frequency data and the initial model. When the starting model is far from the real model, FWI tends to converge to a local minimum. Based on a scale separation of the model (into the background model and reflectivity model), reflection waveform inversion (RWI) can separate out the tomography term in the conventional FWI kernel and invert for the long-wavelength components of the velocity model by smearing the reflected wave residuals along the transmission (or “rabbit-ear”) paths. We have developed a new elastic RWI method to build the P- and S-wave velocity macromodels. Our method exploits a traveltime-based misfit function to highlight the contribution of tomography terms in the sensitivity kernels and a sensitivity kernel decomposition scheme based on the P- and S-wave separation to suppress the high-wavenumber artifacts caused by the crosstalk of different wave modes. Numerical examples reveal that the gradients of the background models become sufficiently smooth owing to the decomposition of sensitivity kernels and the traveltime-based misfit function. We implement our elastic RWI in an alternating way. At each loop, the reflectivity model is generated by elastic least-squares reverse time migration, and then the background model is updated using the separated traveltime kernels. Our RWI method has been successfully applied in synthetic and real reflection seismic data. Inversion results demonstrate that the proposed method can retrieve preferable low-wavenumber components of the P- and S-wave velocity models, which are reliable to serve as a starting model for conventional elastic FWI. Also, our method with a two-stage inversion workflow, first updating the P-wave velocity using the PP kernels and then updating the S-wave velocity using the PS kernels, is feasible and robust even when P- and S-wave velocities have different structures.

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