Acoustic wavefield imaging using the energy norm

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. S151-S163 ◽  
Author(s):  
Daniel Rocha ◽  
Nicolay Tanushev ◽  
Paul Sava
Keyword(s):  
Waveform Inversion ◽  
Spatial Location ◽  
Energy Norm ◽  
Inner Product ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Imaging Condition ◽  
Time Migration ◽  
Head Waves ◽  
Complex Models

Wavefield energy can be measured by the so-called energy norm. We have extended the concept of “norm” to obtain the energy inner product between two related wavefields. Considering an imaging condition as an inner product between the source and receiver wavefields at each spatial location, we have developed a new imaging condition that represents the total reflection energy. Investigating this imaging condition further, we have found that it accounts for wavefield directionality in space time. Based on the directionality discrimination provided by this imaging condition, we have applied it to attenuate backscattering artifacts in reverse time migration (RTM). This imaging condition can be designed not only to attenuate backscattering artifacts, but also to attenuate any selected reflection angle. By exploiting the flexibility of this imaging condition for attenuating certain angles, we have developed a procedure to preserve the type of events that propagate along the same path, i.e., backscattered, diving, and head waves, leading to a suitable application for full-waveform- inversion (FWI). This application involves filtering the FWI gradient to preserve the tomographic term (waves propagating in the same path) and attenuate the migration term (reflections) of the gradient. We have developed the energy imaging condition applications for RTM and FWI using numerical experiments in simple (horizontal reflector) and complex models (Sigsbee and Marmousi).

Isotropic elastic wavefield imaging using the energy norm

Geophysics ◽  
2016 ◽  
Vol 81 (4) ◽  
pp. S207-S219 ◽  
Author(s):  
Daniel Rocha ◽  
Nicolay Tanushev ◽  
Paul Sava
Keyword(s):  
Normal Incidence ◽  
Spatial Location ◽  
Energy Norm ◽  
Inner Product ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Elastic Wave Equation ◽  
Imaging Condition ◽  
Time Migration

From the elastic-wave equation and the energy conservation principle, we have derived an energy norm that is applicable to imaging with elastic wavefields. Extending the concept of the norm to an inner product enables us to compare two related wavefields. For example, the inner product of source and receiver wavefields at each spatial location leads to an imaging condition. This new imaging condition outputs a single image representing the total reflection energy, and it contains individual terms related to the kinetic and potential energy (strain energy) from both extrapolated wavefields. An advantage of the proposed imaging condition compared with alternatives is that it does not suffer from polarity reversal at normal incidence, as do conventional images obtained using converted waves. Our imaging condition also accounted for the directionality of the wavefields in space and time. Based on this information, we have modified the imaging condition for attenuation of backscattering artifacts in elastic reverse time migration images. We performed numerical experiments that revealed the improved quality of the energy images compared with their conventional counterparts and the effectiveness of the imaging condition in attenuating backscattering artifacts even in media characterized by high spatial variability.

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.

scholarly journals Elastic-wave reverse-time migration based on decoupled elastic-wave equations and inner-product imaging condition

2016 ◽  
Vol 13 (6) ◽  
pp. 953-963 ◽  
Author(s):  
Peng Yong ◽  
Jianping Huang ◽  
Zhenchun Li ◽  
Wenyuan Liao ◽  
Luping Qu ◽  
...  
Keyword(s):  
Elastic Wave ◽  
Wave Equations ◽  
Inner Product ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Imaging Condition ◽  
Time Migration ◽  
Elastic Wave Equations

Wavelet-domain reverse time migration image enhancement using inversion-based imaging condition

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. S401-S409
Author(s):  
Hong Liang ◽  
Houzhu Zhang
Keyword(s):  
Wave Equation ◽  
Weighting Function ◽  
Time Derivative ◽  
Spatial Gradient ◽  
Wavelet Domain ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Imaging Condition ◽  
Time Migration ◽  
Head Waves

Reverse time migration (RTM) is implemented by solving the two-way wave equation using recorded data as boundary conditions. The full wave equation can simulate wave propagation in all directions; thus, RTM has no dip limitations and is capable of imaging complex structures. Because wavefields are allowed to travel in all directions, the source and receiver wavefields can be scattered back from strong velocity contrasts. The crosscorrelation of head waves, diving waves, and backscattered waves along a raypath can lead to strong artifacts in the RTM image. These artifacts degrade the final image quality. An inversion-based imaging condition that computes the weighted sum of a time derivative image and a spatial gradient image can significantly reduce the RTM artifacts. Based on the multiscale directional selectivity property of the wavelet transform, we have developed a new method to compute the weighting function for the inversion-based imaging condition in the wavelet domain. The unique property of this approach is that the weighting function depends on the spatial locations, wavenumber, and local directions. This multidimensional property allows us to selectively remove the RTM image artifacts while preserving useful energy. We determine the effectiveness of our method for attenuating RTM artifacts using synthetic examples.

Efficient wavefield reconstruction at half the Nyquist rate

Geophysics ◽  
2019 ◽  
Vol 84 (5) ◽  
pp. A43-A46
Author(s):  
Ali Gholami ◽  
Alan Richardson ◽  
Toktam Zand ◽  
Alison Malcolm
Keyword(s):  
Waveform Inversion ◽  
Time Derivative ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
Simple Method ◽  
Imaging Condition ◽  
Full Waveform ◽  
Time Migration ◽  
Adjoint State ◽  
Nyquist Rate

We have developed a simple method to halve the memory required to store the forward wavefield for the imaging condition in adjoint-state methods such as reverse time migration and full-waveform inversion. It stores the wavefield at only half the Nyquist rate, and it uses the wave equation to calculate the second time derivative, allowing approximate reconstruction of the forward wavefield at the required Nyquist rate. We have determined that the method produces an image with the Marmousi model that is not visibly different compared to a traditional (full Nyquist) implementation.

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.

Reducing artifacts of elastic reverse time migration by the deprimary technique

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. S569-S577 ◽  
Author(s):  
Yang Zhao ◽  
Houzhu Zhang ◽  
Jidong Yang ◽  
Tong Fei
Keyword(s):  
Complex Function ◽  
Noise Removal ◽  
Pure Mode ◽  
Reverse Time ◽  
Reverse Time Migration ◽  
S Wave ◽  
Imaging Condition ◽  
Velocity Models ◽  
Time Migration ◽  
Wavefield Decomposition

Using the two-way elastic-wave equation, elastic reverse time migration (ERTM) is superior to acoustic RTM because ERTM can handle mode conversions and S-wave propagations in complex realistic subsurface. However, ERTM results may not only contain classical backscattering noises, but they may also suffer from false images associated with primary P- and S-wave reflections along their nonphysical paths. These false images are produced by specific wave paths in migration velocity models in the presence of sharp interfaces or strong velocity contrasts. We have addressed these issues explicitly by introducing a primary noise removal strategy into ERTM, in which the up- and downgoing waves are efficiently separated from the pure-mode vector P- and S-wavefields during source- and receiver-side wavefield extrapolation. Specifically, we investigate a new method of vector wavefield decomposition, which allows us to produce the same phases and amplitudes for the separated P- and S-wavefields as those of the input elastic wavefields. A complex function involved with the Hilbert transform is used in up- and downgoing wavefield decomposition. Our approach is cost effective and avoids the large storage of wavefield snapshots that is required by the conventional wavefield separation technique. A modified dot-product imaging condition is proposed to produce multicomponent PP-, PS-, SP-, and SS-images. We apply our imaging condition to two synthetic models, and we demonstrate the improvement on the image quality of ERTM.

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