Target-oriented wave-equation inversion: regularization in the reflection angle
A complex velocity model produces shadow zones in an image due to focusing and defocusing of the seismic waves, and limited recording geometry. These shadow zones contain weak signal masked by artifacts. To recover the real signal, and reduce artifacts is necessary to go beyond migration. One option is to use a wave-equation target-oriented inversion scheme that explicitly computes the least squares inversion Hessian. The wave-equation target-oriented inversion has a big null space. It seeks to form an image where there is lack or very little data information. In this situation is where a priori information in the form of model regularization can help to stabilize the results. One choice for regularization, that makes physical sense, is to force the inverse image to be smooth with the reflection angle. It works by spreading the image from well illuminated to poorly illuminated reflection angles. In order to impose this smoothness constraint I implemented a chain of the subsurface-offset Hessian and a slant-stack (reflection angle to subsurface-offset) operator. Results on the Sigsbee synthetic model show that the inversion regularized in the reflection angle reduces the effect of the uneven illumination not only in the angle gathers but also in the stack image.
Plane-wave migration in tilted coordinates
Most existing one-way wave-equation migration algorithms have difficulty in imaging steep dips in a medium with strong lateral velocity variation. We propose a new one-way wave-equation-based migration, called ``plane-wave migration in tilted coordinates.'' The surface data are converted to plane-wave source data by slant-stacking processing, and each resulting plane-wave source dataset is migrated independently in a tilted coordinate system with an extrapolation direction determined by the source plane-wave direction at the surface. For most waves illuminating steeply dipping reflectors, the extrapolation direction is closer to their propagation direction in the tilted coordinates. Therefore, plane-wave migration in tilted coordinates can correctly image steeply dipping reflectors, even by applying one-way extrapolators. In a well-chosen tilted coordinate system, waves that overturn in conventional vertical Cartesian coordinates do not overturn in the new coordinate system. Using plane-wave migration in tilted coordinates, we can image overturned energy with much lower cost compared to reverse-time migration.
Angle domain common image gathers for steep reflectors
Downward continuation migration cannot provide reliable angle domain common image gathers (CIGs) for steeply dipping reflectors, because it cannot handle most waves that illuminate steep reflectors. Also there is a severe stretch in conventional horizontal subsurface offset at steep reflectors. Both reverse-time migration and plane-wave migration in tilted coordinates solve these two problems and provide robust angle domain CIGs for steeply dipping reflectors. A test on the BP velocity benchmark dataset shows that both migration methods generate robust angle domain CIGs that are comparable. When the migration velocity is not correct, the angle domain CIGs from both migration methods show useful moveout information for velocity estimation.
Phase unwrapping of angle-domain common image gathers
In this paper we adapt a phase unwrapping algorithm to estimate the depth shift in Angle-Domain Common Image Gathers (ADCIGs). We show how to set up a linear system of equations tailored to the seismic case and how to solve it by minimizing an measure via iterations of weighted least-squares problems. For this procedure a meaningful choice of initial weights is crucial. We propose to unwrap jointly several angle gathers and show that this can overcome sampling deficiencies in the angle domain, such as those that come from processing a limited number of subsurface offsets for angle-gather generation.
Angle-domain parameters computed via weighted slant-stack
Angle-domain common image gathers (ADCIGs), created from downward-continuation or reverse time migration, can provide useful lithological and velocity information (Prucha et al., 1999). In geologically complex areas, poor illumination causes undesirable kinematic effects and amplitude variations along the angle axis (Valenciano, 2006). The subsurface-offset-to-angle transformation consists of a radial trace transform in the Fourier space with some regularization in the angle direction (Sava and Fomel, 2000) or slant-stack in the physical space plus an additional transformation from offset ray-parameter to reflection angle (Prucha et al., 1999). The regularization, to some extent, can diminish the amplitude variation caused by poor illumination. The more accurate solution to the illumination problem, however, is achieved by computing a regularized least-squares inverse image (Clapp, 2005) rather than the simply the adjoint (migration). The inverse image problem can be solved either by computing the Hessian implicitly (Clapp, 2005) or explicitly Valenciano (2006), preferentially, in the reflection-angle domain or, without any physically meaningful regularization direction, in the subsurface-offset domain.
Ignoring density in waveform inversion
We study the effectiveness of velocity-only time-domain waveform inversion for inverting synthetic data modeled with both velocity and density contrasts. We present a detailed review of the Born approximation for the constant-density acoustic wave equation and its application to the inversion of velocity models for seismic reflection data. We create synthetic models with both constant and variable density and compare the effectiveness of velocity-only waveform inversion in each case. Results from this simple test suggest that density contrasts can hamper the reconstruction of velocity perturbations.
Exact seismic velocities for TI media and extended Thomsen formulas for stronger anisotropies
I explore a different type of approximation to the exact anisotropic wave velocities as a function of incidence angle in transversely isotropic (TI) media. This formulation extends the Thomsen weak anisotropy approach to stronger deviations from isotropy without significantly affecting the simplicity of the equations. One easily recognized improvement is that the extreme value of the quasi-SV-wave speed is located near the correct incidence angle , rather than always being at the position 45, which universally holds for Thomsen's approximation - although is actually never correct for any TI anisotropic medium. Also, the magnitudes of all the wave speeds are typically (although there may be some exceptions depending on the actual angular location of the extreme value) more closely approximated for all values of the incidence angle. Furthermore, the value of a special angle (which is close to the location of the extreme and also required by the new formulas) can be deduced from the same data that are normally used in the weak anisotropy data analysis. All the main technical results presented are independent of the physical source of the anisotropy. To illustrate the use of the results obtained, two examples are presented based on systems having vertical fractures. The first set of model fractures has their axes of symmetry randomly oriented in the horizontal plane. Such a system is then isotropic in the horizontal plane and, thus, exhibits vertical transverse isotropic (VTI) symmetry. The second set of fractures also has its axes of symmetry in the horizontal plane, but (it is assumed) these axes are aligned so that the system exhibits horizontal transverse isotropic (HTI) symmetry. Both types of systems, as well as any other TI medium (whether due to fractures or layering or other physical causes) are more accurately treated with the new wave speed formulation.
Prediction error filters to enhance differences
Prediction Error Filters (PEFs) capture the covariance of a dataset. In this paper I use PEFs to quantify and highlight difference between two volumes. A series of PEFs are estimated on one volume and then applied to a second. The resulting hypercube is an indicator of where, and how much, two volumes differ.
Accelerating seismic computations using customized number representations on FPGAs
Field Programmable Gate Arrays (FPGA) offer significant potential speedups over conventional microprocessors for some applications. For downward continued migration, complex math and Fast Fourier Transforms (FFT) are the dominant cost. Convolution is the dominant cost for reverse time migration. We implement these core algorithms on a FPGA and show speedups ranging from 5 to 15, including the transfer time to and from the processors. We consider methods to further speed up these migration algorithms.
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