Elastic wavefield separation based on the Helmholtz decomposition

Geophysics ◽  
2017 ◽  
Vol 82 (2) ◽  
pp. S173-S183 ◽  
Author(s):  
Hejun Zhu
Keyword(s):  
Vector Fields ◽  
Helmholtz Decomposition ◽  
Elastic Media ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
S Waves ◽  
Poisson Solver ◽  
Time Migration ◽  
Imaging Conditions

Divergence and curl operators used for the decomposition of P- and S-wave modes in elastic reverse time migration (RTM) change the amplitudes, units, and phases of extrapolated wavefields. I separate the P- and S-waves in elastic media based on the Helmholtz decomposition. The decomposed wavefields based on this approach have the same amplitudes, units, and phases as the extrapolated wavefields. To avoid expensive multidimensional integrals in the Helmholtz decomposition, I introduce a fast Poisson solver to efficiently solve the vector Poisson’s equation. This fast algorithm allows us to reduce computational complexity from [Formula: see text] to [Formula: see text], where [Formula: see text] is the total number of grid points. Because the decomposed P- and S-waves are vector fields, I use vector imaging conditions to construct PP-, PS-, SS-, and SP-images. Several 2D numerical examples demonstrate that this approach allows us to accurately and efficiently decompose P- and S-waves in elastic media. In addition, elastic RTM images based on the vector imaging conditions have better quality and avoid polarity reversal in comparison with images based on the divergence and curl separation or direct component-by-component crosscorrelation.

SP- and SS-imaging for 3D elastic reverse time migration

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. A1-A6 ◽  
Author(s):  
Xufei Gong ◽  
Qizhen Du ◽  
Qiang Zhao
Keyword(s):  
Three Dimensional ◽  
Scalar Wave ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Sh Waves ◽  
S Waves ◽  
Time Migration ◽  
Underlying Principle ◽  
Imaging Conditions

Three-dimensional elastic reverse time migration has been confronted with the problem of generating scalar images with vector S-waves. The underlying principle for solving this problem is to convert the vector S-waves into scalars. Previous methods were mainly focused on PS-imaging, but they usually cannot work properly on SP- and SS-cases. The complexity of SP- and SS-imaging arises from the fact that the incident S-wave has unpredictable relationship with the raypath plane. We have suggested that S-wave should be treated separately as SV- and SH-waves, which keep predictable relationships with the raypath plane. First, the elastic wavefield is separated into P- and S-waves using the Helmholtz decomposition. Then, we evaluate the normal direction of the raypath plane at each imaging grid. Next, we separate the vector S-wave obtained with curl operator into SH- and SV-waves, both of which are scalars. Finally, correlation imaging conditions are implemented to those scalar wave modes to produce scalar SV-P, SV-SV, and SH-SH images.

Multidirectional-vector-based elastic reverse time migration and angle-domain common-image gathers with approximate wavefield decomposition of P- and S-waves

Geophysics ◽  
2018 ◽  
Vol 83 (1) ◽  
pp. S57-S79 ◽  
Author(s):  
Chen Tang ◽  
George A. McMechan
Keyword(s):  
Wave Mode ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
S Waves ◽  
Time Migration ◽  
Mode Separation ◽  
Reflection Image ◽  
Wavefield Decomposition ◽  
Imaging Conditions

Elastic reverse time migration (E-RTM) has limitations when the migration velocities contain strong contrasts. First, the traditional scheme of P/S-wave mode separation is based on Helmholtz’s equations, which ignore the conversion between P- and S-waves at the current separation time. Thus, it contains an implicit assumption of the constant shear modulus and requires smoothing the heterogeneous model to approximately satisfy a locally constant condition. Second, the vector-based imaging condition needs to use the reflection-image normal, and it also cannot give the correct polarity of the PP image in all possible conditions. Third, the angle-domain common-image gathers (ADCIGs) calculated using the Poynting vectors (PVs) do not consider the wave interferences that happen at each reflector. Therefore, smooth models are often used for E-RTM. We relax this condition by proposing an improved data flow that involves three new contributions. The first contribution is an improved system of P/S-wave mode separation that considers the converted wave generated at the current time, and thus it does not require the constant-shear-modulus assumption. The second contribution is the new elastic imaging conditions based on multidirectional vectors; they can give the correct image polarity in all possible conditions without knowledge of the reflection-image normal. The third contribution is two methods to calculate multidirectional propagation vectors (PRVs) for RTM images and ADCIGs: One is the elastic multidirectional PV, and the other uses the sign of wavenumber-over-frequency ([Formula: see text]) ratio obtained from an amplitude-preserved approximate-propagation-angle-based wavefield decomposition to convert the particle velocities into multidirectional PRVs. The robustness of the improved data flow is determined by several 2D numerical examples. Extension of the schemes into 3D and amplitude-preserved imaging conditions is also possible.

Imaging conditions for elastic reverse time migration

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. S95-S111 ◽  
Author(s):  
Wei Zhang ◽  
Ying Shi
Keyword(s):  
Decomposition Method ◽  
Inner Product ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Imaging Condition ◽  
Subsurface Structures ◽  
Time Migration ◽  
Imaging Conditions ◽  
Wave Modes

Elastic reverse time migration (RTM) has the ability to retrieve accurately migrated images of complex subsurface structures by imaging the multicomponent seismic data. However, the imaging condition applied in elastic RTM significantly influences the quality of the migrated images. We evaluated three kinds of imaging conditions in elastic RTM. The first kind of imaging condition involves the crosscorrelation between the Cartesian components of the particle-velocity wavefields to yield migrated images of subsurface structures. An alternative crosscorrelation imaging condition between the separated pure wave modes obtained by a Helmholtz-like decomposition method could produce reflectivity images with explicit physical meaning and fewer crosstalk artifacts. A drawback of this approach, though, was that the polarity reversal of the separated S-wave could cause destructive interference in the converted-wave image after stacking over multiple shots. Unlike the conventional decomposition method, the elastic wavefields can also be decomposed in the vector domain using the decoupled elastic wave equation, which preserves the amplitude and phase information of the original elastic wavefields. We have developed an inner-product imaging condition to match the vector-separated P- and S-wave modes to obtain scalar reflectivity images of the subsurface. Moreover, an auxiliary P-wave stress image can supplement the elastic imaging. Using synthetic examples with a layered model, the Marmousi 2 model, and a fault model, we determined that the inner-product imaging condition has prominent advantages over the other two imaging conditions and generates images with preserved amplitude and phase attributes.

2D and 3D elastic wavefield vector decomposition in the wavenumber domain for VTI media

Geophysics ◽  
2010 ◽  
Vol 75 (3) ◽  
pp. D13-D26 ◽  
Author(s):  
Qunshan Zhang ◽  
George A. McMechan
Keyword(s):  
Vector Fields ◽  
Model Potential ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Polarization Distribution ◽  
Practical Applications ◽  
S Waves ◽  
Wavenumber Domain ◽  
Time Migration ◽  
Vti Media

A pragmatic decomposition of a vector wavefield into P- and S-waves is based on the Helmholtz theory and the Christoffel equation. It is applicable to VTI media when the plane-wave polarization is continuous in the vicinity of a given wavenumber and is uniquely defined by that wavenumber, except for the kiss singularities on the VTI symmetry axis. Unlike divergence and curl, which separate the wavefield into a scalar and a vector field, the decomposed P- and S-wavefields are both vector fields, with correct amplitude, phase, and physical units. If the vector components of decomposed wavefields are added, they reconstruct those of the original input wavefield. Wavefield propagation in any portions of a VTI medium that have the same polarization distribution (i.e., the same eigenvector) in the wavenumber domain have the same decomposition operators and can be recon-structed with a single 3D Fourier transform for each operator (e.g., one for P-waves and one for S-waves).This applies to isotropic wavefields and to VTI anisotropic wavefields, if the polarization distribution is constant, regardless of changes in the velocity. Because the anisotropic phase polarization is local, not global, the wavefield decomposition for inhomogeneous anisotropic media needs to be done separately for each region that has a different polarization distribution. The complete decomposed vector wavefield is constructed by combining the P-, SV-, and SH-wavefields in each region into the corresponding composite P-, SV-, and SH-wavefields that span the model. Potential practical applications include extraction of separate images for different wave types in prestack reverse time migration, inversion, or migration velocity analysis, and calculation of wave-propagation directions for common-angle gathers.

Local vertical seismic profiling (VSP) elastic reverse-time migration and migration resolution: Salt-flank imaging with transmitted P-to-S waves

Geophysics ◽  
2010 ◽  
Vol 75 (2) ◽  
pp. S35-S49 ◽  
Author(s):  
Xiang Xiao ◽  
W. Scott Leaney
Keyword(s):  
Velocity Model ◽  
Staggered Grid ◽  
Image Point ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
S Waves ◽  
Seismic Profiling ◽  
Vertical Seismic Profiling ◽  
Time Migration

To avoid the defocusing effects of propagating waves through salt and overburden with an inaccurate overburden velocity model, we introduce a vertical seismic profiling (VSP) local elastic reverse-time-migration (RTM) method for salt-flank imaging by transmitted P-to-S waves. This method back-projects the transmitted PS waves using a local velocity model around the well until they are in phase with the back-projected PP waves at the salt boundaries. The merits of this method are that it does not require the complex overburden and salt-body velocities and it automatically accounts for source-side statics. In addition, the method accounts for kinematic and dynamic effects, including anisotropy, absorption, and all other unknown rock effects outside of this lo-cal subsalt velocity model. Numerical tests on an elastic salt model and offset 2D VSP data in the Gulf of Mexico, using a finite-difference time-domain staggered-grid RTM scheme, partly demonstrate the effectiveness of this method over interferometry PS-PP transmission migration and local acoustic RTM. Our method separates elastic wavefields to vector P- and S-wave velocity components at the trial image point and achieves better resolution than local acoustic RTM and interferometric transmission migration. The analytical formulas of migration resolution for local acoustic and elastic RTM show that the migration illumination is limited by data frequency and receiver aperture, and the spatial resolution is lower than standard poststack and prestack migration. This new method can image salt flanks as well as subsalt reflectors.

Isotropic elastic reverse time migration using the phase- and amplitude-corrected vector P- and S-wavefields

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. S489-S503 ◽  
Author(s):  
Jidong Yang ◽  
Hejun Zhu ◽  
Wenlong Wang ◽  
Yang Zhao ◽  
Houzhu Zhang
Keyword(s):  
Helmholtz Decomposition ◽  
Pure Mode ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Imaging Condition ◽  
Computational Costs ◽  
Time Migration ◽  
Wavefield Separation ◽  
Dot Product

In elastic reverse time migration (RTM), wavefield separation is an important step to remove crosstalk artifacts and improve imaging quality. State-of-the-art techniques for wavefield separation in isotropic elastic media include using the Helmholtz decomposition and introducing an auxiliary wave equation. Although these two approaches produce pure-mode vector wavefields with correct amplitudes, phases, and physical units, their computational costs are still high under current computational capability, especially for 3D large-scale problems. Based on the P- and S-wave dispersion relations, we have developed an efficient wavefield separation strategy for elastic RTM. Instead of solving a vector Poisson’s equation in the Helmholtz decomposition, we modify the phases of source wavelet as well as multicomponent records and scale the amplitudes of extrapolated wavefields with the squares of P- and S-wave velocities. This operation allows us to produce vector P- and S-wavefields with the same phases and amplitudes as the input coupled wavefields while significantly reducing computational costs. With the separated vector wavefields, we implemented a modified dot-product imaging condition for elastic RTM. In comparison with the previously proposed dot-product imaging condition, this modified imaging condition enables us to eliminate the effects of multiplication with a cosine function and hence produces migrated images with accurate amplitudes. Several 2D and 3D numerical examples are used to demonstrate the feasibility and robustness of our method for imaging complex subsurface structures.

A wavefield-separation-based elastic least-squares reverse time migration

Geophysics ◽  
2018 ◽  
Vol 83 (3) ◽  
pp. S279-S297 ◽  
Author(s):  
Bingluo Gu ◽  
Zhenchun Li ◽  
Jianguang Han
Keyword(s):  
Least Squares ◽  
Synthetic Data ◽  
P Wave ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
S Waves ◽  
Numerical Tests ◽  
Time Migration

Elastic least-squares reverse time migration (ELSRTM) has the potential to provide improved subsurface reflectivity estimation. Compared with elastic RTM (ERTM), ELSRTM can produce images with higher spatial resolution, more balanced amplitudes, and fewer artifacts. However, the crosstalk between P- and S-waves can significantly degrade the imaging quality of ELSRTM. We have developed an ELSRTM method to suppress the crosstalk artifacts. This method includes three crucial points. The first is that the forward and backward wavefields are extrapolated based on the separated elastic velocity-stress equation of P- and S-waves. The second is that the separated vector P- and S-wave residuals are migrated to form reflectivity images of Lamé constants [Formula: see text] and [Formula: see text] independently. The third is that the reflectivity images of [Formula: see text] and [Formula: see text] are obtained by the vector P-wave wavefields achieved in the backward extrapolation of the separated vector P-wave residuals and the vector S-wave wavefields achieved in the backward extrapolation of the separated vector S-wave residuals, respectively. Numerical tests with synthetic data demonstrate that our ELSRTM method can produce images free of crosstalk artifacts. Compared with ELSRTM based on the coupled wavefields, our ELSRTM method has better convergence and higher accuracy.

Vector-wave-based elastic reverse time migration of ocean-bottom 4C seismic data

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. S333-S343 ◽  
Author(s):  
Pengfei Yu ◽  
Jianhua Geng ◽  
Jiqiang Ma
Keyword(s):  
Seismic Data ◽  
Ocean Bottom ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Vector Wave ◽  
S Waves ◽  
Time Migration ◽  
Multicomponent Seismic ◽  
Wavefield Extrapolation

The acoustic-elastic coupled equation (AECE) has several advantages when compared with conventional scalar-wave-based elastic reverse time migration (ERTM) methods used to image ocean-bottom multicomponent seismic data. In particular, vector-wave-based ERTM requires vectorial P- and S-waves on the source and receiver sides, but these cannot be directly obtained from wavefield extrapolation using AECE. Therefore, we have developed a P- and S-wave vector decomposition (VD) approach within AECE; this approach enables the deduction of a novel VD-based AECE, from which vectorial P- and S-waves can be obtained directly via wavefield extrapolation. We are also able to derive a new formulation suitable for vector-wave-based ERTM of ocean-bottom multicomponent seismic data that can generate a phase-preserved PS-image. Three synthetic examples illustrate the validity and effectiveness of our new method.

Elastic least-squares reverse time migration of steeply dipping structures using prismatic reflections

Geophysics ◽  
2022 ◽  
pp. 1-130
Author(s):  
Zheng Wu ◽  
Yuzhu Liu ◽  
Jizhong Yang
Keyword(s):  
Least Squares ◽  
Seismic Data ◽  
Oil And Gas ◽  
Normal Equation ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Time Migration ◽  
Multicomponent Seismic ◽  
Imaging Conditions

The migration of prismatic reflections can be used to delineate steeply dipping structures, which is crucial for oil and gas exploration and production. Elastic least-squares reverse time migration (ELSRTM), which considers the effects of elastic wave propagation, can be used to obtain reasonable subsurface reflectivity estimations and interpret multicomponent seismic data. In most cases, we can only obtain a smooth migration model. Thus, conventional ELSRTM, which is based on the first-order Born approximation, considers only primary reflections and cannot resolve steeply dipping structures. To address this issue, we develop an ELSRTM framework, called Pris-ELSRTM, which can jointly image primary and prismatic reflections in multicomponent seismic data. When Pris-ELSRTM is directly applied to multicomponent records, near-vertical structures can be resolved. However, the application of imaging conditions established for prismatic reflections to primary reflections destabilizes the process and leads to severe contamination of the results. Therefore, we further improve the Pris-ELSRTM framework by separating prismatic reflections from recorded multicomponent data. By removing artificial imaging conditions from the normal equation, primary and prismatic reflections can be imaged based on unique imaging conditions. The results of synthetic tests and field data applications demonstrate that the improved Pris-ELSRTM framework produces high-quality images of steeply dipping P- and S-wave velocity structures. However, it is difficult to delineate steep density structures because of the insensitivity of the density to prismatic reflections.

Vector-based elastic reverse time migration based on scalar imaging condition

Geophysics ◽  
2017 ◽  
Vol 82 (2) ◽  
pp. S111-S127 ◽  
Author(s):  
Qizhen Du ◽  
ChengFeng Guo ◽  
Qiang Zhao ◽  
Xufei Gong ◽  
Chengxiang Wang ◽  
...  
Keyword(s):  
Alternative Methods ◽  
Polarity Reversal ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Converted Wave ◽  
Imaging Condition ◽  
Time Migration ◽  
Wavefield Separation ◽  
Wave Modes

The scalar images (PP, PS, SP, and SS) of elastic reverse time migration (ERTM) can be generated by applying an imaging condition as crosscorrelation of pure wave modes. In conventional ERTM, Helmholtz decomposition is commonly applied in wavefield separation, which leads to a polarity reversal problem in converted-wave images because of the opposite polarity distributions of the S-wavefields. Polarity reversal of the converted-wave image will cause destructive interference when stacking over multiple shots. Besides, in the 3D case, the curl calculation generates a vector S-wave, which makes it impossible to produce scalar PS, SP, and SS images with the crosscorrelation imaging condition. We evaluate a vector-based ERTM (VB-ERTM) method to address these problems. In VB-ERTM, an amplitude-preserved wavefield separation method based on decoupled elastic wave equation is exploited to obtain the pure wave modes. The output separated wavefields are both vectorial. To obtain the scalar images, the scalar imaging condition in which the scalar product of two vector wavefields with source-normalized illumination is exploited to produce scalar images instead of correlating Cartesian components or magnitude of the vector P- and S-wave modes. Compared with alternative methods for correcting the polarity reversal of PS and SP images, our ERTM solution is more stable and simple. Besides these four scalar images, the VB-ERTM method generates another PP-mode image by using the auxiliary stress wavefields. Several 2D and 3D numerical examples are evaluated to demonstrate the potential of our ERTM method.

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