Four-component Data Research Articles

Exact two-component Hamiltonians for relativistic quantum chemistry: Two-electron picture-change corrections made simple.

Based on self-consistent field (SCF) atomic mean-field (amf) quantities, we present two simple yet computationally efficient and numerically accurate matrix-algebraic approaches to correct both scalar-relativistic and spin-orbit two-electron picture-change effects (PCEs) arising within an exact two-component (X2C) Hamiltonian framework. Both approaches, dubbed amfX2C and e(xtended)amfX2C, allow us to uniquely tailor PCE corrections to mean-field models, viz. Hartree-Fock or Kohn-Sham DFT, in the latter case also avoiding the need for a point-wise calculation of exchange-correlation PCE corrections. We assess the numerical performance of these PCE correction models on spinor energies of group 18 (closed-shell) and group 16 (open-shell) diatomic molecules, achieving a consistent ≈10-5 Hartree accuracy compared to reference four-component data. Additional tests include SCF calculations of molecular properties such as absolute contact density and contact density shifts in copernicium fluoride compounds (CnFn, n = 2,4,6), as well as equation-of-motion coupled-cluster calculations of x-ray core-ionization energies of 5d- and 6d-containing molecules, where we observe an excellent agreement with reference data. To conclude, we are confident that our (e)amfX2C PCE correction models constitute a fundamental milestone toward a universal and reliable relativistic two-component quantum-chemical approach, maintaining the accuracy of the parent four-component one at a fraction of its computational cost.

Elastic Least-Squares Reverse-Time Migration Based on a Modified Acoustic-Elastic Coupled Equation for OBS Four-Component Data

Elastic least-squares reverse-time migration (ELSRTM) yields subsurface high-resolution elastic images from multicomponent seismic data. For ocean bottom seismic (OBS) acquisitions, both pressure and displacement components are measured using four-component (4C) modern receivers. However, these 4C data cannot be effectively simulated and migrated using conventional ELSRTM approaches due to the limitation of the standard elastic wave equation. For this reason, we propose a new ELSRTM method based on a modified acoustic-elastic coupled (AEC) equation. Owing to this equation, either the prediction or the migration of OBS 4C data can be implemented using a one-time wavefield simulation. Incorporation with a perturbation imaging condition, the proposed method can invert for P- and S-wave velocity and density and generate amplitude-preserving images. Besides, it can provide subsurface impedance reflectivity models by stacking the obtained multi-parameter images. In the least-squares migration, a preconditioned conjugate gradient algorithm is implemented based on a multi-parameter diagonal Hessian. Numerical experiments on an uncorrelated layer model and a portion of the Southern Yellow China sea model can demonstrate the effectiveness of this method on improving imaging resolution and accelerating convergence rate.

Application of four-component dipole shear reflection imaging to interpret the geological structure around a deviated well

Acoustic reflection imaging in deep water wells is a new application scope for offshore hydrocarbon exploration. Two-dimensional (2D) geological structure images can be obtained away from a one-dimensional (1D) borehole using single-well acoustic reflection imaging. Based on the directivity of dipole source and four-component dipole data, one can achieve the azimuth detection and the three-dimensional (3D) structural information around the wellbore can be obtained. We first perform matrix rotation on the field four-component data. Then, a series of processing steps are applied to the rotated dipole data to obtain the reflector image. According to the above dipole shear-wave imaging principle, we used four-component cross-dipole logging data from a deviated well in the South China Sea to image geological structures within 50 m of a deviated well, which can delineate the structural configuration and determine its orientation. The configuration of near-borehole bedding boundaries and fault structures from shear-wave imaging results agrees with those from the Inline and Xline seismic profiles of the study area. In addition, the configuration and orientation of the fault structure images are consistent with regional stress maps and the results of the borehole stress anisotropy analysis. Furthermore, the dip azimuth of the bedding boundary images was determined using borehole wall resistivity data. Results of this study indicate that integrating borehole acoustic reflection with seismic imaging not only fi lls the gap between the two measurement scales but also accurately delineates geological structures in the borehole vicinity.

Density Functional Calculations of Electron Paramagnetic Resonance g- and Hyperfine-Coupling Tensors Using the Exact Two-Component (X2C) Transformation and Efficient Approximations to the Two-Electron Spin-Orbit Terms.

A two-component quasirelativistic density functional theory implementation of the computation of hyperfine and g-tensors at exact two-component (X2C) and Douglas-Kroll-Hess method (DKH) levels in the Turbomole code is reported and tested for a series of smaller 3d1, 4d1, and 5d1 complexes, as well as for some larger 5d7 Ir and Pt systems in comparison with earlier four-component matrix-Dirac-Kohn-Sham results. A main emphasis is placed on efficient approximations to the two-electron spin-orbit contributions, comparing an existing implementation of two variants of Boettger's "scaled nuclear spin-orbit" (SNSO) approximation in the code with a newly implemented atomic mean-field spin-orbit (AMFSO) approximation. The different variants perform overall comparably well with the four-component data. The AMFSO approximation has the added advantage of being able to include the spin-other-orbit contributions arising from the Gaunt term of relativistic electron-electron interactions. These are of comparably larger importance for the 3d complexes than for their heavier homologues. The excellent agreement between X2C and four-component electron paramagnetic resonance parameter results provides the opportunity to treat large systems efficiently and accurately with the computationally more expedient two-component quasirelativistic methodology.

A ray casting method for the computation of the area of feasible solutions for multicomponent systems: Theory, applications and FACPACK-implementation

Multivariate curve resolution methods suffer from the non-uniqueness of the solutions. The set of possible nonnegative solutions can be represented by the so-called Area of Feasible Solutions (AFS). The AFS for an s-component system is a bounded (s−1)-dimensional set. The numerical computation and the geometric construction of the AFS is well understood for two- and three-component systems but gets much more complicated for systems with four or even more components.This work introduces a new and robust ray casting method for the computation of the AFS for general s-component systems. The algorithm shoots rays from the origin and records the intersections of these rays with the AFS. The ray casting method is computationally fast, stable with respect to noise and is able to detect the various possible shapes of the AFS sets. The easily implementable algorithm is tested for various three- and four-component data sets.

Fully Relativistic Calculations of Faraday and Nuclear Spin-Induced Optical Rotation in Xenon.

Nuclear spin-induced optical rotation (NSOR) arising from the Faraday effect may constitute an advantageous novel method for the detection of nuclear magnetization. We present first-principles nonrelativistic and relativistic, two- and four-component, basis-set limit calculations of this phenomenon for xenon. It is observed that only by utilization of relativistic methods may one qualitatively reproduce experimental liquid-state NSOR data. Relativistic effects lower the results by 50% as compared to nonrelativistic values. Indeed, relativistic Hartree-Fock calculations at the four-component or exact two-component (X2C) level account for the discrepancy between experimental results and earlier nonrelativistic theory. The nuclear magnetic shielding constant of traditional nuclear magnetic resonance as well as the Verdet constant parametrizing optical rotation due to an external magnetic field were also calculated. A comparison between results obtained using Hartree-Fock and density-functional theory methods at relativistic and nonrelativistic levels, as well as coupled cluster methods at the nonrelativistic level, was carried out. Completeness-optimized basis sets were employed throughout, for the first time in fully relativistic calculations. Full relativity decreases the Verdet constant by 4%. X2C theory decreases the absolute value of NSOR by 10-20% as compared to the four-component data, while for Verdet constants, the results are only slightly smaller than the fully relativistic values. For both properties, two-component calculations decrease the computational time by roughly 90%. Density-functional methods yield substantially larger values of NSOR than the Hartree-Fock theory or experiments. Intermolecular interactions are found to decrease NSOR and, hence, compensate for the electron correlation effect.

A decade of tomography

Over the past 10 years, ray-based postmigration grid tomography has become the standard model-building tool for seismic depth imaging. While the basics of the method have remained unchanged since the late 1990s, the problems it solves have changed dramatically. This evolution has been driven by exploration demands and enabled by computer power. There are three main areas of change. First, standard model resolution has increased from a few thousand meters to a few hundred meters. This order of magnitude improvement may be attributed to both high-quality, complex residual-moveout data picked as densely as [Formula: see text] to [Formula: see text] vertically and horizontally, and to a strategy of working down from long-wavelength to short-wavelength solutions. Second, more and more seismic data sets are being acquired along multiple azimuths, for improved illumination and multiple suppression. High-resolution velocity tomography must solve for all azimuths simultaneously, to prevent short-wavelength velocity heterogeneity from being mistaken for azimuthal anisotropy. Third, there has been a shift from predominantly isotropic to predominantly anisotropic models, both VTI and TTI. With four-component data, anisotropic grid tomography can be used to build models that tie PZ and PS images in depth.

Four-Component Data for Imaging, Lithology Prediction, and Fluid Discrimination

This article is a synopsis of paper OTC 11907, "The Use of Four-Component Data for Imaging, Lithology Prediction, and Fluid Discrimination: Three Case Studies," by Eivind Fromyr and Santi Randazzo, PGS Reservoir Services Inc., originally presented at the 2000 Offshore Technology Conference, Houston, 1-4 May.

Analysis of Four-Component Data for AVO Effects

This article is a synopsis of paper OTC 10984, "Analysis of 4C Data for AVO Effects at Teal South, Eugene Island 354," G.W. Purnell, B.J. Nolte, P.M. Krail, and D.A. Ebrom, Texaco, originally presented at the 1999 Offshore Technology Conference, Houston, 3-6 May.

Linear matrix operations for multicomponent seismic processing

SUMMARY Multicomponent seismic data contain overlapping information on the polarization states of distinct body-wave modes, due to the physical process of excitation, propagation and recording. This geometric redundancy should be exploited to provide an accurate separation and estimation of the wavefield attributes in order to understand the medium properly. This may be achieved using linear transforms, originally developed for separating split shear waves in four-component seismic data. These transforms separate the principal time-series components of the wavefield from the ray-path geometry and the orientation of the source and geophone axes for a uniform medium: they are deterministic and can be easily implemented. Here we reformulate the linear transforms by introducing simple geometry and medium-independent matrix operators. Although for ‘ideal’ experiments the technique may offer nothing new to the estimation of polarization that eigenanalysis cannot offer, nevertheless the formulation avoids the need to consult mathematical libraries and is useful in the interpretation of the wavefield when various inevitable acquisition-related errors dominate. Some typical problems in processing four-component data, such as the interpretation of data matrix asymmetry due to misorientations of the acquisition components and non-orthogonal polarizations for the wave components, may be easily treated and identified using a common framework with this condensed matrix form. In addition, the operation is extended to similar geometric problems in six- and nine-component data. Synthetic and field nine-component data examples are presented to illustrate the application of the matrix operations.

An empirical comparison of resolution methods for three-way arrays

In chemometrics applications it is common to resolve a trilinear array by solving a generalized eigenvalue problem, rather than by employing the iterative PARAFAC algorithm commonly used by psychometricians. Although an eigenanalysis-based procedure works perfectly in the absence of noise, it is not guaranteed to yield least squares resolutions when noise is present. The PARAFAC algorithm on the other hand is guaranteed to reduce the residual sum of squares at each iteration. In this paper we propose synthesizing the two methods by using the resolution generated by eigenanalysis as starting values for the iterative PARAFAC algorithm. We find for simulated four-component data at moderate noise levels that following an eigenanalysis resolution with PARAFAC frequently leads to significant improvement in the quality of the resolution.

A method for obtaining multicomponent diffusion coefficients directly from Rayleigh and Gouy fringe position data

A nonlinear least-squares (NLLS) method for analyzing three-component Rayleigh fringe position data for diffusion, described briefly in previous papers, is extended here to four-component data. By a simple transformation of equations for the Gouy method, an analogous NLLS procedure can be applied directly to three- and four-component Gouy fringe position data. The only complication is that the Gouy NLLS is subjected to a condition equation. Results are given for the free-diffusion case, but the procedure can be extended to other boundary conditions.

Compartmental Efflux Analysis: An Evaluation of the Technique and Its Limitations

Efflux analysis is an established tool for characterizing the exchange properties of multicomponent systems. In this report, we have simulated several three- and four-compartment systems with error-free and imperfect data, the errors being designed to mimic actual, nonbiological variability in isotope efflux studies. The data sets were analyzed using computerized nonlinear regression techniques to identify the important aspects of actual experimental design (uptake times, efflux collection schedules, and total efflux times), and to consider the possibility that a properly designed and executed experiment might fail to resolve compartmentation correctly. The results showed that for any of the systems simulated, including those with error-free four-component data, a reasonable three-component fit was obtainable. Resolution of the additional compartment was not always possible. In correctly resolved systems, failure to estimate the correct decay constants was common, especially when the half-times were separated by less than an order of magnitude. We conclude that efflux analysis, by itself, lacks the power to provide reliable information about multicompartment systems.

A Fourier method for the analysis of exponential decay curves

A method based on the Fourier convolution theorem is developed for the analysis of data composed of random noise, plus an unknown constant "base line," plus a sum of (or an integral over a continuous spectrum of) exponential decay functions. The Fourier method's usual serious practical limitation of needing high accuracy data over a very wide range is eliminated by the introduction of convergence parameters and a Gaussian taper window. A computer program is described for the analysis of discrete spectra, where the data involves only a sum of exponentials. The program is completely automatic in that the only necessary inputs are the raw data (not necessarily in equal intervals of time); no potentially biased initial guesses concerning either the number or the values of the components are needed. The outputs include the number of components, the amplitudes and time constants together with their estimated errors, and a spectral plot of the solution. The limiting resolving power of the method is studied by analyzing a wide range of simulated two-, three-, and four-component data. The results seem to indicate that the method is applicable over a considerably wider range of conditions than nonlinear least squares or the method of moments.