Don Pham

Antileakage Fourier transform for seismic data regularization

Seismic data regularization, which spatially transforms irregularly sampled acquired data to regularly sampled data, is a long-standing problem in seismic data processing. Data regularization can be implemented using Fourier theory by using a method that estimates the spatial frequency content on an irregularly sampled grid. The data can then be reconstructed on any desired grid. Difficulties arise from the nonorthogonality of the global Fourier basis functions on an irregular grid, which results in the problem of “spectral leakage”: energy from one Fourier coefficient leaks onto others. We investigate the nonorthogonality of the Fourier basis on an irregularly sampled grid and propose a technique called “antileakage Fourier transform” to overcome the spectral leakage. In the antileakage Fourier transform, we first solve for the most energetic Fourier coefficient, assuming that it causes the most severe leakage. To attenuate all aliases and the leakage of this component onto other Fourier coefficients, the data component corresponding to this most energetic Fourier coefficient is subtracted from the original input on the irregular grid. We then use this new input to solve for the next Fourier coefficient, repeating the procedure until all Fourier coefficients are estimated. This procedure is equivalent to “reorthogonalizing” the global Fourier basis on an irregularly sampled grid. We demonstrate the robustness and effectiveness of this technique with successful applications to both synthetic and real data examples.

Tomographic residual curvature analysis: The process and its components

The image quality of 3D pre-stack depth migration in areas of complex geology depends strongly on the accuracy of the velocity model. Velocity updating by seismic tomography, in the form of either traveltime tomography or residual curvature analysis (RCA), has become an important component of the depth imaging process. In this paper, we describe the components of tomography, discussing RCA in detail. These components include: building of the tomographic updating equations; regularization in both data and model spaces; and application of the least-squares solver. Numerical examples show that the RCA algorithm works well, producing velocity models that improve the quality of depth-migrated images over models produced by vertical updating schemes.

Tomographic velocity analysis in strongly anisotropic TTI media

Traveltime derivatives with respect to the unknown anisotropy parameters play an important role in anisotropic tomography. Zhou et al. (2003) derived the analytical traveltime derivatives for weakly anisotropic TTI media (transversely isotropic media with arbitrary axis of symmetry). In this abstract, we will further extend that derivation into strongly anisotropic TTI media. We also present simplified version of Snell’s law and traveltime perturbations with respect to the reflector movement in 2.5D TTI media. A synthetic example and a real data example show the effectiveness of our proposed algorithm.

On the orthogonality of anti-leakage Fourier transform based seismic trace interpolation

Summary Seismic data regularization, which aims to estimate the seismic traces on a spatially regular grid using the a cquired irregular sampled data, is an interpolation/extrapolation problem. Sampling theory offers the basic conditions for all the seismic data regularization implementations. In sampling theory, Fourier transform plays a crucial role in the analysis of the reconstruction/interpolation basis (interpolant); it estimates the frequency components in frequency/wave-number domain, and its inverse transform creates the seismic data on the desired regular grid. Difficulties arise from the non-orthogonality of the global Fourier basis on an irregular grid, which results in the energy from one frequency component leaks onto others. This well-known phenomenon is called “spectral leakage”. The updated Fourier transform: Anti-leakage Fourier transform (ALFT) offers to overcome the above mentioned difficulties. It estimates the spatial frequency content on a n irregularly sampled grid with significantly reduced frequency leakage. In this paper, we investigate the properties of ALFT and give an insight on how it works. The interpolants are numerically calculated and analyzed in detail. The orthogonality condition of the interpolants is discussed, which demonstrates that the ALFT data reconstruction meets the two most important interpolation conditions (e.g. orthogonal condition and unity condition). With the amplitude analysis on interpolants, the stability of ALFT algorithm is also addressed.

3-D Tomographic Velocity Analysis In Transversely Isotropic Media

Most current anisotropic velocity analysis deals with nonhyperbolic normal-moveout (NMO), with application to prestack time processing. In contrast, we present a general algorithm for 3-D velocity updating with tomographic velocity analysis, with application to 3-D prestack depth imaging. First, we outline the key components of tomographic velocity analysis in transversely isotropic media with arbitrary axis of symmetry (TTI media). Second, we derive an explicit representation of traveltime derivatives in weak transversely isotropic media and present the corresponding analytic formulae. Finally, we demonstrate the effectiveness of this procedure with a simple synthetic data example.

Global solution to water column statics: A new approach to an old problem

We develop a novel procedure to solve the long-standing problem of water column statics. Our procedure consists of automatic picking of the relative static shift between the sail lines, and solving this problem as a global nonlinear inversion problem. Like most inversion problems, the solutions are not unique. So, we introduce a priori constraints to limit the final solution. The constraints require the least movement of traces when statics are derived. Furthermore, we implement a hybrid l/l norm in our algorithm for a robust inversion; this norm is relatively insensitive to large residuals, which may occur as the result of incorrect picks in the automatic picking. Results on synthetic and real data sets show that our method is robust and powerful.

Velocity Update Via Joint Velocity Inversion For Anisotropic Depth Migration

We present a new velocity update procedure for anisotropic P-P and P-S prestack depth migration in the transversely isotropic media with vertical symmetry (VTI). The anisotropic parameters are estimated by jointly inverting velocity from prestack P-P and P-S depth image gathers. The estimation consists of three steps. The first step is to create depth consistent image gather via joint velocity inversion in depth domain to estimate the vertical depth, vertical Pand S-wave velocities. This step assumes that the corresponding reflections from P-P and P-S have been identified. The second step is to perform isotropic depth migration by scanning the focusing velocities from the depth image gathers. The third step is to compute anisotropic parameters based on the results of the first and second steps. Integrating anisotropic parameters with depth consistent velocity model from joint velocity inversion completes the anisotropic prestack depth migration procedure. Our method allows for the calculation of anisotropic parameters directly in the depth domain. It is more stable and easy to tie with well log data for reservoir interpretation. The estimation of the vertical Pand S-wave velocities are useful for anisotropic processing when well log data is not available. The results from synthetic and real data show that the estimation method is a practical approach in the processing of multicomponent data.

Key Aspects of 3D Reflection Traveltime Tomography

In addition to derivative computation, we need to pick both traveltimes and local time dips from prestack common shot and common receiver gathers. These picks are, then, mapped by ray tracing into horizontal focusing errors, which are used as the input for the inversion. For complex models, this mapping may introduce additional errors in the inversion. As a consequence, we further develop a datapreconditioning scheme to ensure the stability of the inversion.

Coherent noise velocity analysis by phase‐matching plus p‐omega domain wave field transformation and its application in ground roll attenuation

Ground roll is one of the major types of coherent noise in land data. Attenuation of ground roll must be done before any further processing of desired signals such as P-wave reflections. Effective removal methods must take into count the following properties of ground roll: frequency dispersion, high amplitude and low velocity. In this paper, a method is designed to estimate the velocity of coherent noises: a combination of phase-matching and (p, ) domain wave field transformation. Both synthetic and real examples demonstrate that this method is easy to implement and can be used to effectively estimate the dispersion relation of coherent noises of different modes and separate these noises (e.g. ground rolls) from other useful signals.