High Frequency Assumption Research Articles

Uniqueness in the magnetic shaping problem without high frequency assumption

In this work, we study an inverse source problem arising from the magnetic casting process. Specifically, we discuss the mathematical model without the high frequency assumption on the current in inductors. We derive an associated transmission problem and provide a proof of uniqueness by using the Bedrosian identity.

WKB solutions to the quasi 1-D acoustic wave equation in ducts with non-uniform cross-section and inhomogeneous mean flow properties – Acoustic field and combustion instability

Linear modal analysis based on the spatio-temporal harmonic representation of acoustic fluctuations is a well-known reduced order method to predict combustion instability frequencies. However, a limitation of this method is that it is only applicable to domains with homogeneous mean properties. The Wentzel-Kramers-Brillouin (WKB) method facilitates the development of approximate analytical solutions to the one-dimensional (1-D) acoustic wave equation in domains with inhomogeneous mean properties. The “classical” form of the WKB solution to the wave equation is based on the assumption of high frequency (or, short wavelength), and is applicable to uniform cross-section ducts with a mean temperature gradient, but no mean flow. Subsequently, the WKB method was extended to include uniform mean velocity as well. In the current study, we derive a WKB-type solution to the generalized Helmholtz equation in quasi 1-D ducts with non-uniform cross-sectional areas and inhomogeneities in mean flow properties such as the velocity, temperature, density, and pressure. The WKB solution is applicable under the assumptions of high frequency and slowly varying mean properties. In deriving this solution, a novel approach has been developed to relate the first and second spatial derivatives of the axial velocity fluctuations to the pressure fluctuations (and its derivatives).A second WKB-type solution (referred to as the WKB2 solution) is also obtained by relaxing the slowly-varying mean property assumption, but retaining the high frequency approximation. Thirdly, the Helmholtz equation for pressure is solved numerically without invoking the two WKB assumptions. The WKB, WKB2 and numerical solutions are compared for a number of cases involving combinations of non-uniform duct areas and axially varying mean properties. Finally, we apply the WKB solution to predict the longitudinal combustion instabilities in a 1-D combustor characterized by a sudden expansion, with uniform mean flow in each of the two ducts comprising the combustor, and a linear temperature profile downstream of a compact, planar flame.

Frequency-dependent traveltime tomography using fat rays: application to near-surface seismic imaging

Frequency-dependent traveltime tomography does not rely on the high frequency assumption made in classical ray-based tomography. By incorporating the effects of velocity structures in the first Fresnel volume around the central ray, it offers a more realistic and accurate representation of the actual physics of seismic wave propagation and thus, enhanced imaging of near-surface structures is expected. The objective of this work was to apply frequency-dependent first arrival traveltime tomography to surface seismic data that were acquired for exploration scale and near-surface seismic imaging. We adapted a fat ray tomography algorithm from global-earth seismology that calculates the Fresnel volumes based on source and receiver (adjoint source) traveltime fields. The fat ray tomography algorithm was tested on synthetic model data that mimics the dimensions of two field data sets. The field data sets are presented as two case studies where fat ray tomography was applied for near-surface seismic imaging. The data set of the first case study was recorded for high-resolution near-surface imaging of a Quaternary valley (profile length<1km); the second data set was acquired for hydrocarbon search (profile length>10km). All results of fat ray tomography are compared against the results of classical ray-based tomography. We show that fat ray tomography can provide enhanced tomograms and that it is possible to recover more information on the subsurface when compared to ray tomography. However, model assessment based on the column sum of the Jacobian matrix revealed that especially the deep parts of the structure in the fat ray tomograms might not be adequately covered by fat rays. Furthermore, the performance of the fat ray tomography depends on the chosen input frequency in relation to the scale of the seismic survey. Synthetic data testing revealed that the best results were obtained when the frequency was chosen to correspond to an approximate wavelength-to-target depth ratio of 0.1.

Generalized Drude scattering rate from the memory function formalism: an independent verification of the Sharapov-Carbotte result

An explicit perturbative computation of the Mori's memory function was performed by G\"otze and W\"olfle (GW) to calculate Generalized Drude scattering (GDS) rate for the case of electron-impurity and electron-phonon scattering in metals by assuming constant electronic density of states at the Fermi energy. In the present investigation, we go beyond this assumption and extend the GW formalism to the case in which there is a gap around the Fermi surface in electron density of states. The resulting GDS is compared with a recent one by Sharapov and Carbotte (SC) obtained through a different route. We find good agreement between the two at finite frequencies. However, we find discrepancies in the dc scattering rate. These are due to a crucial assumption made in SC namely $\omega >> \vert \Sigma(\epsilon+\omega) - \Sigma^{*}(\epsilon)\vert$. No such high frequency assumption is made in the memory function based technique.

Seismic wavefield propagation in 2D anisotropic media: Ray theory versus wave-equation simulation

Despite the ray theory that is based on the high frequency assumption of the elastic wave-equation, the ray theory and the wave-equation simulation methods should be mutually proof of each other and hence jointly developed, but in fact parallel independent progressively. For this reason, in this paper we try an alternative way to mutually verify and test the computational accuracy and the solution correctness of both the ray theory (the multistage irregular shortest-path method) and the wave-equation simulation method (both the staggered finite difference method and the pseudo-spectral method) in anisotropic VTI and TTI media . Through the analysis and comparison of wavefield snapshot, common source gather profile and synthetic seismogram , it is able not only to verify the accuracy and correctness of each of the methods at least for kinematic features, but also to thoroughly understand the kinematic and dynamic features of the wave propagation in anisotropic media . The results show that both the staggered finite difference method and the pseudo-spectral method are able to yield the same results even for complex anisotropic media (such as a fault model); the multistage irregular shortest-path method is capable of predicting similar kinematic features as the wave-equation simulation method does, which can be used to mutually test each other for methodology accuracy and solution correctness. In addition, with the aid of the ray tracing results, it is easy to identify the multi-phases (or multiples) in the wavefield snapshot, common source point gather seismic section and synthetic seismogram predicted by the wave-equation simulation method, which is a key issue for later seismic application. • Wavefield simulation in 2D anisotropic media using a pseudo-spectral method • Multiply arrival tracking in 2D anisotropic media using a multistage ISPM method • Comparison tests were conducted for the above two methods. • It is possible to mutually verify accuracy and correctness of each method above.

Simultaneous Inversion for Velocity and Reflector Geometry Using Multi-phase Fresnel Volume Rays

Traditional ray tomography methods based on the high frequency assumption are sometimes unable to obtain a high resolution tomographic picture due to a deficient coverage of ray paths in real applications, especially for low velocity anomalous regions. In contrast, finite-frequency ray theory is more suitable for handling real seismic propagation problems because the travel time depends not only on the velocity distribution along a central ray (or traditional geometric ray), but also on the velocity values within a region (referred to as the first Fresnel Volume) which incorporates the central ray. In this study, we develop an algorithm to calculate multi-phase Fresnel Volume finite-frequency rays, and then present an inversion method to simultaneous invert for both velocity and reflector geometry by using these multi-phase Fresnel Volume finite-frequency rays. Using synthetic data examples, we compare the reconstructions of the velocity field and the reflector orientation using the Fresnel Volume ray tomographic methods and the traditional ray tomography approach. Results show that the former is advantageous over the latter, especially when the ray density is relatively low. An additional benefit of the Fresnel Volume finite-frequency ray tomographic method is that it can start with a low frequency to capture the coarse velocity structure, thereby mitigating the local minimum trapping problem, and then be tuned to a high frequency for delineating the fine velocity structure.

An Application of the Level Set Method to Underwater Acoustic Propagation

An algorithm for computing wavefronts, based on the high frequency approximation to the wave equation, is presented. This technique applies the level set method to underwater acoustic wavefront propagation in the time domain. The level set method allows for computation of the acoustic phase function using established numerical techniques to solve a first order transport equation to a desired order of accuracy. Traditional methods for solving the eikonal equation directly on a fixed grid limit one to only the first arrivals, so these approaches are not useful when multi-path propagation is present. Applying the level set model to the problem allows for the time domain computation of the phase function on a fixed grid, without having to restrict to first arrival times. The implementation presented has no restrictions on range dependence or direction of travel, and offers improved efficiency over solving the full wave equation which under the high frequency assumption requires a large number of grid points to resolve the highly oscillatory solutions. Boundary conditions are discussed, and an approach is suggested for producing good results in the presence of boundary reflections. An efficient method to compute the amplitude from the level set method solutions is also presented. Comparisons to analytical solutions are presented where available, and numerical results are validated by comparing results with exact solutions where available, a full wave equation solver, and with wavefronts extracted from ray tracing software.

An Application of the Level Set Method to Underwater Acoustic Propagation

An algorithm for computing wavefronts, based on the high frequency approximation to the wave equation, is presented. This technique applies the level set method to underwater acoustic wavefront propagation in the time domain. The level set method allows for computation of the acoustic phase function using established numerical techniques to solve a first order transport equation to a desired order of accuracy. Traditional methods for solving the eikonal equation directly on a fixed grid limit one to only the first arrivals, so these approaches are not useful when multi-path propagation is present. Applying the level set model to the problem allows for the time domain computation of the phase function on a fixed grid, without having to restrict to first arrival times. The implementation presented has no restrictions on range dependence or direction of travel, and offers improved efficiency over solving the full wave equation which under the high frequency assumption requires a large number of grid points to resolve the highly oscillatory solutions. Boundary conditions are discussed, and an approach is suggested for producing good results in the presence of boundary reflections. An efficient method to compute the amplitude from the level set method solutions is also presented. Comparisons to analytical solutions are presented where available, and numerical results are validated by comparing results with exact solutions where available, a full wave equation solver, and with wavefronts extracted from ray tracing software.

Estimation of the Expected Discounted Penalty Function for Levy Insurance Risks

We consider a generalized risk process which consists of a subordinator plus a spectrally negative Levy process. Our interest is to estimate the expected discounted penalty function (EDPF) from a set of data which is practical in the insurance framework. We construct an empirical type estimator of the Laplace transform of the EDPF, and obtain it by a regularized Laplace inversion. The asymptotic behavior of the estimator under a high frequency assumption is investigated.

Estimation of the expected discounted penalty function for Lévy insurance risks

We consider a generalized risk process which consists of a subordinator plus a spectrally negative Levy process. Our interest is to estimate the expected discounted penalty function (EDPF) from a set of data which is practical in the insurance framework. We construct an empirical type estimator of the Laplace transform of the EDPF and obtain it by a regularized Laplace inversion. The asymptotic behavior of the estimator under a high frequency assumption is investigated.

NUMERICAL METHODS FOR WAVE SCATTERING PHENOMENA BY MEANS OF DIFFERENT BOUNDARY INTEGRAL FORMULATIONS

Different numerical approaches for the physical phenomena of scattering waves from an obstacle are presented. They are based on different integral formulations. Fluid structure interaction effects are numerically treatable as well. We use the Boundary Element Method (BEM) in different approaches because the inherently satisfied Sommerfeld radiation condition makes sure that no reflecting waves from boundaries at infinity occur. One of the biggest disadvantages of numerical methods like BEM is the fact that they have difficulties with handling the high frequency range. For the high frequency range approximations like the Kirchhoff–Helmholtz integral equation have to be used. With varying assumptions of the reflecting behavior of the structure different approaches for the higher frequency range are obtained, where the explicit solving of a system of equations is not necessary. Another high frequency approach is the plane wave approximation which is compared with the Kirchhoff approach of the first kind. Additionally a modified Kirchhoff approach is introduced. Because the incident pressure on the scatterer's surface is known the integral is evaluated analytically on triangular patches. The discretization is no longer frequency dependent and the size of the patches only depends on the curvature of the structure. Large planar parts can be discretized with one element only. This leads to a substantial advantage in terms of calculation time over the traditional Kirchhoff approach. Like the traditional approach this procedure is valid under the assumption of high frequency or far field conditions.

Rapid Method for Acoustic Wave‐Equation WTW Inversion of Crosshole Seismic Data

AbstractWTW is a hybrid inversion method of wave‐equation traveltime inversion (WT) and waveform inversion. Wave function is used to calculate the traveltime and derivative (perturbation of traveltime with respect to velocity) in WT. Unlike traditional ray‐based travetime inversion, WT has many advantages. No ray tracing or traveltime picking and no high frequency assumption is necessary and good result can be got while starting model is far from real model. Comparing with waveform inversion, WT has low resolution. Complementary to WT, waveform inversion has high resolution, but easily sticks in local minima when starting model is far from real model. So WTW is a good hybrid inversion method. Traditional WTW is to first reconstruct the smooth characters of model with WT inversion being primary and then reconstruct the detailed characters of model with waveform inversion being primary. But the computation time and memory using of this method are twice of that of WT inversion or waveform inversion. So a simplified method is used actually. That is to first use WT singly and then waveform inversion singly. Advantages and disadvantages of two inverse methods are not sufficiently complement for iterations with simplified method, this affects the inverse result. This paper proposes a new method to implement WTW and computation time and memory using are same with that of WT or waveform inversion. Model calculation shows the new method has better convergence.

Prestack wave-equation depth migration in VTI media

Wave-equation prestack depth migration (PSDM) naturally handles multi-arrivals, is not limited by the high frequency assumption of Kirchhoff-based PSDM and can, therefore, produce higher quality images in areas of complex geology. In addition, it has been shown that anisotropy, commonly encountered in petroleum exploration, affects image focusing and positioning. Therefore incorporating anisotropy in wave equation PSDM will produce a superior image!

Numerical simulations of depth‐dependent anisotropy and frequency‐dependent wave propagation effects

A numerical investigation of the effects of shear wave splitting for vertical propagation in a smoothly varying anisotropic medium is presented. Through forward modeling, we predict the olivine lattice preferred orientation (LPO) developed in the oceanic upper mantle in response to the absolute plate motion (APM). We consider the effect of a change in APM similar to the one that presumably caused the kink in the Emperor‐Hawaii seamount island chain in the north Pacific. This results in an oblique orientation between lithospheric and asthenospheric anisotropy. Numerical simulations of shear wave propagation are used to estimate the characteristics of shear‐wave splitting. Ray theory does not account for coupling between shear waves in the depth‐dependent anisotropic medium due to the implicit assumption of high frequency. A forward propagator technique for calculating waveforms and splitting parameters is used to assess frequency‐dependent effects. The results show that ray theory is valid for estimating the splitting only for frequencies above 1 Hz. At frequencies more realistic for SKS propagation, apparent splitting parameters exhibit a π/2 dependence on the incoming shear wave polarization (back azimuth). For certain back azimuth ranges, shear wave splitting is very frequency dependent with apparent delay times ranging from 1 to 4 s and apparent fast polarization directions changing rapidly by up to 80°. Thus stacking of shear wave splitting measurements for largely different initial polarizations and frequencies should be avoided. Depth‐dependent anisotropy implies that shear wave splitting analyses will be sensitive to filtering. Anisotropic depth variations cannot be resolved unambiguously from splitting observations at relatively long periods (&gt;5 s). It is not possible, for instance, to discriminate between smooth and abrupt transitions separating the anisotropic regions. Shorter‐period waveforms provide further information on the fine structure of anisotropic depth variations. A comparison between splitting calculations and observations from Hawaii suggests a divergent past APM direction or may indicate an alternative mechanism responsible for the lithospheric anisotropy.

Calculation of Added Resistance in Oblique Waves

The formula for the added resistance of a ship in oblique waves is simplified by the assumption of high frequency of encounter. Numerical computations by the simplified formula are carried out with respect to the Series 60, CB=0.70 model at various heading angles to the wave. The results of computations are compared with results of experiments by Yamanouchi and Ando, and fairly good agreement between them is observed when the ship is moving against waves. This fact seems to prove the feasibility of the above formula as a practical method of the theoretical prediction of added resistance in oblique waves.