Sverre Brandsberg-Dahl

Focusing in dip and AVA compensation on scattering‐angle/azimuth common image gathers

Common image gathers (CIGs) in the offset and surface azimuth domain are used extensively in migration velocity analysis and amplitude variation with offset (AVO) studies. If the geology is complex and the ray field becomes multipathed, the quality of the CIGs deteriorates. To overcome these problems, the CIGs are generated as a function of scattering angle and azimuth at the image point. The CIGs are generated using an algorithm based on the inverse generalized Radon transform (GRT), stacking only over migration dip angles. Including only dips in the vicinity of the geological dip, or focusing in dip, suppresses artifacts in and results in improved signal‐to‐noise ratio on the CIGs.Migration velocity analysis can be based upon the differential semblance criterion. The analysis~is~then carried out by minimizing a functional of the derivative of the CIGs with respect to horizontal coordinates (offset/azimuth or scattering‐angle/azimuth), but AVO/amplitude variation with angle (AVA) effects will degrade the...

The pseudo-analytical method: application of pseudo-Laplacians to acoustic and acoustic anisotropic wave propagation

Summary We generalize the pseudo-spectral method for the acoustic wave equation to create analytical solutions to the constant velocity acoustic wave equation in an arbitrary number of space dimensions. We accomplish this by modifying the Fourier Transform of the Laplacian operator so that it compensates exactly for the error due to the second-order finite-difference time marching scheme used in the conventional pseudo-spectral method. Of more practical interest, we show that this modified or pseudo-Laplacian is a smoothly varying function of the parameters of the acoustic wave equation (velocity most importantly) and thus can be further generalized to create near-analyticallyaccurate solutions for the variable velocity case. We call this new method the pseudo-analytical method. We further show that by applying this approach to the concept of acoustic anisotropic wave propagation, we can create scalar-mode VTI and TTI wave equations that overcome the disadvantages of previously published methods for acoustic anisotropic wave propagation. These methods should be ideal for forward modeling and reverse time migration applications.

Maslov asymptotic extension of Generalized Radon Transform inversion in anisotropic elastic media: a Least-Squares approach

Linearized asymptotic inversion of seismic data is carried out in general anisotropic media. In an anisotropic medium, even if it is homogeneous, the shear waves form (instantaneous) caustics. In the absence of caustics, we formulated the seismic inverse scattering problem via the generalized Radon transform. In the presence of caustics, which are associated with multi-pathing, we have to re-derive the inversion procedure which is now based on the weak formulation of the inverse problem. The key ingredients are the Maslov canonical operators describing the transmission from the image point to the sources and the receivers, and delicate high-dimensional stationary phase analyses. The importance of caustics in practical applications becomes apparent in the analysis of mode converted wave constituents in sedimentary basins.

TTI reverse time migration using the pseudo-analytic method

Reverse time migration (RTM) was first introduced in the early 1980s (Whitmore, 1983), but was seemingly dormant until recent advances in computer hardware helped propel it onto the stage as a powerful depth-imaging method. RTM is now standard for areas where large velocity contrasts and/or steep dips pose a challenge, for instance below salt in the Gulf of Mexico. In recent years, where called for by the data, the migration tool of choice has gone from isotropic RTM to anisotropic RTM. The most common representations of anisotropy in sedimentary rocks are VTI (transverse isotropy with a vertical axis of symmetry) and TTI (tilted transverse isotropy). While isotropic and VTI RTM have become somewhat routine, TTI RTM remains challenging due to the complexity, stability, computational cost, and the difficulty in estimating the anisotropic parameters for TTI media.

3D TTI RTM Using the Pseudo-analytic Method

We use an extension of the pseudo-spectral method, called the pseudo-analytic method, to formulate a 3D TTI reverse time migration scheme. The pseudo-analytic method provides accurate, nearly non-dispersive wave propagation with a simple 2-order time-stepping scheme. The formulation is easily adapted to describe scalar wave propagation in VTI and TTI media. Our efficient TTI RTM scheme is free of the pseudo-shear wave artifacts that are normally associated with VTI and TTI formulations. We show 2D synthetic and 3D field data examples to illustrate the high-quality images that pseudo-analytic RTM produces.

Wave equation migration with attenuation and anisotropy compensation

We introduce a new viscoacoustic Wave Equation Migration (WEM) for media with attenuation. Our solution is based on a Fourier Finite-Difference (FFD) scheme for migration by wavefield continuation. Similarly to the acoustic solution, the viscoacoustic migration consists of three terms: a phase-shift extrapolation, a thin-lens correction, and a finite-differences operation. The viscoacoustic migration is also extended to account for anisotropy (VTI and TTI). The anisotropic effects are incorporated in the migration by using odd and even rational function terms in the finite differences solution. The dispersion relation, in presence of attenuation, includes both real and imaginary terms. While the real part controls the kinematics of the image, the imaginary part recovers the high vertical wave numbers in the seismic image; therefore improving resolution and amplitude balance. The implementation is stable, efficient, and very flexible. In absence of attenuation or anisotropy, the solution reduces to the familiar isotropic acoustic case. Results from synthetic example and two dual sensor field surveys from the North Sea and the Gulf of Mexico demonstrate the importance of incorporating the attenuation effects in isotropic and anisotropic migration algorithms.

Multi‐parameter controlled automatically picking and variable smoothing for tomography with fast 3D beam prestack depth migration

Automatically picking on 3D volumes and fast convergence to variable wavelength are two major challenges in grid-based reflection tomography. This paper discusses a multi-parameter controlled automatically picking method that, when integrated with a fast beam migration, provides reliable picks of primary events while avoiding coherent noises such as multiples. To speed up the convergence of the solution, we use a 3D Gaussian filter as a smoothing operator that enables variable smoothing lengths for different parts of the model. Combining the efficiency and flexibility of the beam migration with the two proposed developments, we provide a robust and economical tool for velocity model building. Field data tests show uplift in image quality when using an updated velocity model after one pass of the iterative solution.

Hybrid tomography based on beam migration

We propose a method of tomography that utilizes the time residual and other attributes associated with each wavelet in beam migration. This time residual is referred to as 3D residual normal moveout (3DRNMO) and is related to error in the velocity model. The spatial dip components of the wavelet with respect to time enable very efficient ray tracing. The proposed method possesses the features of both postmigration tomography and stereotomography. Furthermore, not only reflections but also diffractions and turning waves are included in the tomography. The method is validated using a TTI synthetic data set from a 3D syncline model. The method is also applied to fine tune the velocity model for a marine field data set.

Implicit Wave-equation Migration In TTI Media Using High Order Operators

This paper discusses high order implicit operators for migration by wavefield continuation in media with tilted transverse isotropy (TTI). The operators are built using Pade’s rational series expansion in the wave number domain, which translates to implicit finite-differences schemes in the space domain. They can be part of pure implicit finite-differences or mixed domain migration algorithms, e.g., Fourier finite-differences (FFD). Unlike the isotropic case, the TTI Pade’s expansion is formed by a combination of odd and even order polynomials. A third order approximation to the TTI dispersion relation is shown to be both accurate and efficient. Its numerical implementation requires a pentadiagonal solver instead of the conventional tridiagonal solver used with second and fourth order schemes.

True geometry tomography for velocity model building with applications to WATS seismic data

Tomography has been widely employed for velocity model building. The typical work flow starts with the migration of an initial velocity model. This is followed by picking residual moveouts and then updating the velocity through tomography. The migration process provides common image gathers and a stack. At the tomography stage, a ray tracer is used to trace specular rays from the image points to the surface to set up the system of linear equations for the tomographic inversion by linking valid ray pairs to their corresponding residuals. For narrow azimuth (NAZ) surveys, searching for valid ray pairs is usually limited to a narrow azimuth band. Since neither the gathers nor the stack contain acquisition geometry information, the selected specular ray pairs may not reflect the true ray paths, resulting in inaccurate rays being used in the inversion process. In addition to the problem of “which rays to choose”, we also have the problem of “how many rays to choose”. These problems are even more difficult to handle with acquisition configurations other than NAZ, such as the wide azimuth towed-streamer (WATS) surveys, multi-azimuth (MAZ) surveys, and ocean bottom cable (OBC) surveys. To overcome these problems, we propose a tomography method that incorporates the acquisition geometry information and uses vector offsets to account for both offsets and azimuths. To address the ill-posed nature of the system of equations, we developed an anisotropic Laplacian regularization operator that allows different smoothing along different directions. We validate the method with tests on both synthetic and field data with a WATS geometry.