The complications in anisotropic multi-parameter inversion lie in the trade-off between the different anisotropy parameters. We compute the tomographic waveform sensitivity kernels for a VTI acoustic medium perturbation as a tool to investigate this ambiguity between the different parameters. We use dynamic ray tracing to efficiently handle the expensive computational cost for 3-D anisotropic models. Ray tracing provides also the ray direction information necessary for conditioning the sensitivity kernels to handle anisotropy. The NMO velocity and η parameter kernels showed a maximum sensitivity for diving waves which results in a relevant choice of those parameters in wave equation tomography. The δ parameter kernel showed zero sensitivity; therefore it can serve as a secondary parameter to fit the amplitude in the acoustic anisotropic inversion. Considering the limited penetration depth of diving waves, migration velocity analysis based kernels are introduced to fix the depth ambiguity with reflections and compute sensitivity maps in the deeper parts of the model.
Phase wrapping is one of main causes of the local minima problem in waveform inversion. However, the unwrapping process for 2D phase maps that includes singular points (residues) is complicated and does not guarantee unique solutions. We employ an exponential damping to eliminate the residues in the 2D phase maps, which makes the 2D phase unwrapping process easy and produce a unique solution. A recursive inversion process using the damped unwrapped phase provides an opportunity to invert for smooth background updates first, and higher resolution updates later as we reduce the damping. We also apply a Gaussian filter to the gradient to mitigate the edge artifacts resulting from the narrow shape of the sensitivity kernels at high damping. Numerical examples demonstrate that our unwrapped phase inversion with damping and a Gaussian filter produces good convergent results even for a 3Hz single frequency of Marmousi dataset and with a starting model far from the true model.
Aldawood, Ali; Hoteit, Ibrahim; Turkiyyah, George M.; Zuberi, M. A H; Alkhalifah, Tariq Ali(Proceedings 76th EAGE Conference and Exhibition 2014, EAGE Publications, 2014)[Conference Paper]
Internal multiples deteriorate the quality of the migrated image obtained conventionally by imaging single scattering energy. However, imaging internal multiples properly has the potential to enhance the migrated image because they illuminate zones in the subsurface that are poorly illuminated by single-scattering energy such as nearly vertical faults. Standard migration of these multiples provide subsurface reflectivity distributions with low spatial resolution and migration artifacts due to the limited recording aperture, coarse sources and receivers sampling, and the band-limited nature of the source wavelet. Hence, we apply a linearized least-square inversion scheme to mitigate the effect of the migration artifacts, enhance the spatial resolution, and provide more accurate amplitude information when imaging internal multiples. Application to synthetic data demonstrated the effectiveness of the proposed inversion in imaging a reflector that is poorly illuminated by single-scattering energy. The least-square inversion of doublescattered data helped delineate that reflector with minimal acquisition fingerprint.
Zuberi, M. A H; Alkhalifah, Tariq Ali(Proceedings 76th EAGE Conference and Exhibition 2014, EAGE Publications, 2014)[Conference Paper]
Angle gathers provide an extra dimension to analyze the velocity after migration. Space-shift and time shift-imaging conditions are two methods used to obtain angle gathers, but both are reasonably expensive. By scaling the time-lag axis of the time-shifted images, the computational cost of the time shift imaging condition can be considerably reduced. In imaging and more so Full waveform inversion, frequencydomain Helmholtz solvers are used more often to solve for the wavefields than conventional time domain extrapolators. In such cases, we do not need to extend the image, instead we scale the frequency axis of the frequency domain image to obtain the angle gathers more efficiently. Application on synthetic data demonstrate such features.
Full waveform inversion (FWI) despite it's potential suffers from the ability to converge to the desired solution due to the high nonlinearity of the objective function at conventional seismic frequencies. Even if frequencies necessary for the convergence are available, the high number of iterations required to approach a solution renders FWI as very expensive (especially in 3D). A spectral implementation in which the wavefields are extrapolated and gradients are calculated in the wavenumber domain allows for a cleaner more efficient implementation (no finite difference dispersion errors). In addition, we use not only an up and down going wavefield decomposition of the gradient to access the smooth background update, but also a right and left propagation decomposition to allow us to do that for large dips. To insure that the extracted smooth component of the gradient has the right decent direction, we solve an optimization problem to search for the smoothest component that provides a negative (decent) gradient. Application to the Marmousi model shows that this approach works well with linear increasing initial velocity model and data with frequencies above 2Hz.
Using the reflection imaging process as a source to model reflections for full waveform inversion (FWI), referred to as reflection FWI (RFWI), allows us to update the background component of the model, and avoid using the relatively costly migration velocity analysis (MVA), which usually relies on extended images. However, RFWI requires a good image to represent the current reflectivity, as well as, some effort to obtain good smooth gradients. We develop a spectral implementation of RFWI where the wavefield extrapolations and gradient evaluation are performed in the wavenumber domain, obtaining clean dispersion free and fast extrapolations. The gradient, in this case, yields three terms, two of which provide us with each side of the rabbit ear kernel, and the third, often ignored, provides a normalization of the reflectivity within the kernel, which can be used to obtain a reflectivity free background update. Since the image is imperfect (it is an adjoint, not an inverse), an optimization process for the third term scaling is implemented to achieve the smoothest gradient update. A rare application of RFWI on the reflectivity infested Marmousi model shows some of the potential of the approach.
Prestack depth migration in anisotropic media, especially those that exhibit tilt, can be costly using reverse time migration (RTM). We present two-way spectral extrapolation of prestack exploding reflector modeling and migration (PERM) in acoustic transversely isotropic (TI) media. We construct systematic ways to evaluate phase angles and phase velocities in dip oriented TI (DTI), vertical TI (VTI) and tilted TI (TTI) media. Migration results from the Marmousi VTI model and the BP2007 TTI model show the feasibility of our approach.
Wu, Zedong; Alkhalifah, Tariq Ali(EAGE Workshop on High Performance Computing for Upstream, EAGE Publications, 2014)[Conference Paper]
We have presented a new method that can be dispersion free and unconditionally stable. Thus the computational cost and memory requirement will be reduced a lot. Based on this feature, we have implemented this algorithm on GPU based CUDA for the anisotropic Reverse time migration. There is almost no communication between CPU and GPU. For the prestack wavefield extrapolation, it can combine all the shots together to migration. However, it requires to solve a bigger dimensional problem and more meory which can't fit into one GPU cards. In this situation, we implement it based on domain decomposition method and MPI for distributed memory system.
The acoustic approximation is an efficient alternative to the equations of elastodynamics for modeling Pwave propagation in weakly anisotropic media. We present a stable discontinuous Galerkin (DG) method for solving the acoustic approximation in tilted TI media (acoustic TI approximation). The acoustic TI approximation is considered as a modification of the equations of elastodynamics from which a modified energy is derived. The modified energy is obtained by eliminating the shear stress in the coordinates determined by the tilt angle and finding an energy for the remaining unknowns. This construction is valid if the medium is not elliptically anisotropic, a requirement frequently found in the literature. In the fully discrete setting, the modified energy is also conserved in time the presence of sharp contrasts in material parameters. By construction, the scheme can be coupled to the (fully) acoustic wave equation in the same way as the equations of elastodynamics. Hence, the number of unknowns can be reduced in acoustic regions. Our numerical examples confirm the conservation of energy in the discrete setting and the stability of the scheme.
Integrating migration velocity analysis (MVA) and full waveform inversion (FWI) can help reduce the high nonlinearity of the classic FWI objective function. The combination of inverting for the long and short wavelength components of the velocity model using a dual objective function that is sensitive to both components is still very expensive and have produced mixed results. We develop an approach that includes both components integrated to complement each other. We specifically utilize the image to generate reflections in our synthetic data only when the velocity model is not capable of producing such reflections. As a result, we get the MVA working when we need it, and mitigate it's influence when the velocity model produces accurate reflections (possible first for the low frequencies). Applications to a layered model, as well as, the Marmousi model demonstrate some of the approach features.
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