Infinite Frequency Approximation Research Articles

Seismic Imaging of the Waltham Canyon Fault, California: Comparison of Ray‐Theoretical and Fresnel Volume Prestack Depth Migration

Abstract Near‐vertical faults can be imaged using reflected refractions identified in controlled‐source seismic data. Often theses phases are observed on a few neighboring shot or receiver gathers, resulting in a low‐fold data set. Imaging can be carried out with Kirchhoff prestack depth migration in which migration noise is suppressed by constructive stacking of large amounts of multifold data. Fresnel volume migration can be used for low‐fold data without severe migration noise, as the smearing along isochrones is limited to the first Fresnel zone around the reflection point. We developed a modified Fresnel volume migration technique to enhance imaging of steep faults and to suppress noise and undesired coherent phases. The modifications include target‐oriented filters to separate reflected refractions from steep‐dipping faults and reflections with hyperbolic moveout. Undesired phases like multiple reflections, mode conversions, direct P and S waves, and surface waves are suppressed by these filters. As an alternative approach, we developed a new prestack line‐drawing migration method, which can be considered as a proxy to an infinite frequency approximation of the Fresnel volume migration. The line‐drawing migration is not considering waveform information but requires significantly shorter computational time. Target‐oriented filters were extended by dip filters in the line‐drawing migration method. The migration methods were tested with synthetic data and applied to real data from the Waltham Canyon fault, California. The two techniques are applied best in combination, to design filters and to generate complementary images of steep faults.

Electronic and vibrational linear and nonlinear polarizabilities of Li@C60 and [Li@C60

Electronic and vibrational nuclear relaxation (NR) contributions to the dipole (hyper)polarizabilities of the endohedral fullerene Li@C(60) and its monovalent cation [Li@C(60)](+) are calculated at the (U)B3LYP level. Many results are new, while others differ significantly from those reported previously using more approximate methods. The properties are compared with those of the corresponding hypothetical noninteracting systems with a valence electron transferred from Li to the cage. Whereas the NR contribution to the static linear polarizabilities is small in comparison with the corresponding electronic property, the opposite is true for the static hyperpolarizabilities. A relatively small, but non-negligible, NR contribution to the dc-Pockels effect is obtained in the infinite frequency approximation.

Ocean acoustic tomography

Ocean acoustic tomography (OAT) was proposed in 1979 by Walter Munk and Carl Wunsch as an analogue to x-ray computed axial tomography for the oceans. The oceans are opaque to most electromagnetic radiation, but there is a strong acoustic waveguide, and sound can propagate for 10 Mm and more with distinct multiply-refracted ray paths. Transmitting broadband pulses in the ocean leads to a set of impulsive arrivals at the receiver which characterize the impulse response of the sound channel. The peaks observed at the receiver are assumed to represent the arrival of energy traveling along geometric ray paths. These paths can be distinguished by arrival time, and by arrival angle when a vertical array of receivers is available. Changes in ray arrival time can be used to infer changes in ocean structure. Ray travel time measurements have been a mainstay of long-range acoustic measurements, but the strong sensitivity of ray paths to range-dependent sound speed perturbations makes the ray sampling functions uncertain in real cases. In the ray approximation travel times are sensitive to medium changes only along the corresponding eigenrays. Ray theory is an infinite-frequency approximation, and its eikonal equation has nonlinearities not found in the acoustic wave equation. We build on recent seismology results (kernels for body wave arrivals in the earth) to characterize the kernel for converting sound speed change in the ocean to travel time changes using more complete propagation physics. Wave-theoretic finite frequency kernels may show less sensitivity to small-scale sound speed structure.

High-resolution aquifer characterization using seismic cross-hole tomography: An evaluation experiment in a gravel delta

We performed field experiments at a well-studied aquifer test site to evaluate the geophysical method of seismic cross-hole tomography for high-resolution aquifer characterization. We address the challenge of finding meaningful relationships between seismic velocity and hydraulic conductivity at the field scale for hydraulic conductivity estimation. The resolution of seismic cross-hole data is limited by inversion ambiguities and diffraction effects, which are not accounted for using conventional ray-based inversion algorithms for velocity reconstruction. We investigate the impact of diffraction effects on seismic velocity inversion and implications for hydraulic modelling. Our ray-based seismic tomograms correlate well with sedimentary cores and hydraulic conductivity logs allowing us to interpret a zone model with distinct geophysical/hydraulic zones. We use this zone model to constrain seismic inversions using two approaches for forward modelling: ray tracing (infinite-frequency approximation) and wavefield modelling at a realistic frequency, which accounts for diffraction. We show that using the common ray tracing approach, seismic velocities in low-velocity zones may be significantly overestimated. Based on the available data, we observe a negative linear relationship between logarithmic hydraulic conductivity and seismic velocity. This relationship is used to parameterize hydraulic models based on seismic velocity models. Computer simulations of advective contaminant transport demonstrate that their results are extremely sensitive to an accurate determination of seismic low-velocity zones. We conclude that seismic tomography is indeed very useful for high-resolution aquifer characterization. However, an inversion approach that accounts for diffraction effects, a priori knowledge of the subsurface structure and additional well data should be applied for quantitative interpretations.

Simple finite field method for calculation of static and dynamic vibrational hyperpolarizabilities: Curvature contributions

In the static field limit, the vibrational hyperpolarizability consists of two contributions due to: (1) the shift in the equilibrium geometry (known as nuclear relaxation), and (2) the change in the shape of the potential energy surface (known as curvature). Simple finite field methods have previously been developed for evaluating these static field contributions and also for determining the effect of nuclear relaxation on dynamic vibrational hyperpolarizabilities in the infinite frequency approximation. In this paper the finite field approach is extended to include, within the infinite frequency approximation, the effect of curvature on the major dynamic nonlinear optical processes.

Nuclear relaxation contribution to static and dynamic (infinite frequency approximation) nonlinear optical properties by means of electrical property expansions: Application to HF, CH4, CF4, and SF6

Electrical property derivative expressions are presented for the nuclear relaxation contribution to static and dynamic (infinite frequency approximation) nonlinear optical properties. For CF4 and SF6, as opposed to HF and CH4, a term that is quadratic in the vibrational anharmonicity (and not previously evaluated for any molecule) makes an important contribution to the static second vibrational hyperpolarizability of CF4 and SF6. A comparison between calculated and experimental values for the difference between the (anisotropic) Kerr effect and electric field induced second-harmonic generation shows that, at the Hartree–Fock level, the nuclear relaxation/infinite frequency approximation gives the correct trend (in the series CH4, CF4, SF6) but is of the order of 50% too small.

Analysis of the vibrational, static and dynamic, second hyperpolarizability of five small molecules

Various approximate methods for the calculation of vibrational second hyperpolarizabilities are assessed on the basis of results for CH4, NH3, H2O, HF, and CO2. Three specific variations of the perturbation-theoretic approach are analyzed. These are defined by the types of terms retained in expansions which involve electrical and mechanical anharmonicities. The lowest-order approximation being the double-harmonic one and the next higher one the relaxation method. For the dynamic hyperpolarizabilities, the replacement of the optical frequencies (ω) by ω→∞ (the infinite-frequency approximation) is also explored. On the evidence at hand it is concluded that the relaxation/infinite-frequency method is a satisfactory approximation for the nonlinear optical vibrational hyperpolarizabilities.