Delta Function Approximation Research Articles

Suppressing position approximation imaging artifacts in full waveform modeling and inversion of marine seismic data

Full-waveform inversion (FWI) is an advanced seismic imaging method for determining high resolution velocity models of different scales, such as in subseafloor geo-hazard monitoring and risk analysis. Among all two-way wave-equation based modeling methods, a mixed-grid finite-difference modeling (FDM) in frequency domain has the advantage of low numerical dispersion errors for mesh size up to 1/4 wave-length. However, irregular source and receiver positions situated away from the FDM nodes introduce position approximation error (PAE) as large as half mesh size. When a source or receiver is situated in strong heterogeneity or near a free surface, the PAE induces imaging artifacts in high frequencies. We propose an imaging artifact suppression method for multi-scale FDM and FWI in marine seismic survey. The delta function of the source or receiver position is approximated by a discrete Kaiser-windowed sinc function in unbounded medium. The free-surface effects to sources and receivers are compensated with the mirror source concept. The final coefficients for delta function approximation and wavefield interpolation are spatially confined tensor products. The results using ocean bottom cable and vertical cable survey layouts in synthetic models validate the accuracy and feasibility of this new FDM and FWI strategy. The FWI tests shows that this strategy has reduced small scale artifacts at high frequencies for survey layouts of good and poor illuminations. The extension of our method to 3-D visco-acoustic model or reverse time migration is straightforward.

Modeling of Flow Structure, Bubble Distribution, and Heat Transfer in Polydispersed Turbulent Bubbly Flow Using the Method of Delta Function Approximation

The results of modeling of flow structure, air bubble distribution over the pipe cross section, and heat transfer in a vertical polydispersed gas-liquid flow are presented. Themathematical model is based on the Euler description with allowance for the back effect of bubbles on the averaged characteristics and turbulence of the carrier phase. The polydispersity of two-phase flow is described by the delta approximation method with consideration of bubble break-up and coalescence. The turbulence of the carrier phase is predicted using the Reynolds stress transport equations. The results of the modeling showed good agreement with experimental and numerical data of other works.

Scattering of a short electromagnetic pulse from a Lorentz–Duffing film: Theoretical and numerical analysis

We combine scattering theory, Fourier, traveling wave and asymptotic analyses together with numerical simulations to present interesting and practically useful properties of femtosecond pulse interaction with thin films. The dispersive material is described by a single resonance Lorentz model and its nonlinear extension with a cubic Duffing-type nonlinearity. A key feature of the Lorentz dielectric function is that its real part becomes negative between its zero and its pole, generating a forbidden region. We illustrate numerically the linear interaction of the pulse with the film using both scattering theory and Fourier analysis. Outside this region we show the generation of a sequence of pulses separated by round trips in the Fabry–Perot cavity due to multiple reflections. When the pulse spectrum is inside the forbidden region, we observe total reflection. Near the pole of the dielectric function, we demonstrate the slowing down of the pulse (group velocity tending to zero) in the medium that behaves as a high-Q cavity. We use the combination of analysis and simulations in the linear regime to validate the delta function approximation of the thin layer; this collapses the forbidden region to a single resonant point of the spectrum. We also study the single cycle pulse interaction with a thin film and show three distinct types of reflection: half-pulse, sinusoidal wave train and cosine wavelet. Finally we analyze the influence of a strong nonlinearity and observe that the film switches from reflecting to transparent.

Particle-without-Particle: A Practical Pseudospectral Collocation Method for Linear Partial Differential Equations with Distributional Sources

Partial differential equations with distributional sources---in particular, involving (derivatives of) delta distributions---have become increasingly ubiquitous in numerous areas of physics and applied mathematics. It is often of considerable interest to obtain numerical solutions for such equations, but any singular ("particle"-like) source modeling invariably introduces nontrivial computational obstacles. A common method to circumvent these is through some form of delta function approximation procedure on the computational grid; however, this often carries significant limitations on the efficiency of the numerical convergence rates, or sometimes even the resolvability of the problem at all. In this paper, we present an alternative technique for tackling such equations which avoids the singular behavior entirely: the "Particle-without-Particle" method. Previously introduced in the context of the self-force problem in gravitational physics, the idea is to discretize the computational domain into two (or more) disjoint pseudospectral (Chebyshev-Lobatto) grids such that the "particle" is always at the interface between them; thus, one only needs to solve homogeneous equations in each domain, with the source effectively replaced by jump (boundary) conditions thereon. We prove here that this method yields solutions to any linear PDE the source of which is any linear combination of delta distributions and derivatives thereof supported on a one-dimensional subspace of the problem domain. We then implement it to numerically solve a variety of relevant PDEs: hyperbolic (with applications to neuroscience and acoustics), parabolic (with applications to finance), and elliptic. We generically obtain improved convergence rates relative to typical past implementations relying on delta function approximations.

Heat generation and conduction in PDMS-carbon nanoparticle membranes irradiated with optical fibers

Carbon based materials have recently been used to fabricate highly photo-absorbing surfaces for heat generation purposes. In this report we present experimental measurements of the temperature increments ΔT in polymer (PDMS)-carbon nanopowder membranes irradiated by a light source via an optical fiber. In particular, we analyze experimentally the effects of the optical depth, τλ=βL, on the transient and steady state temperature profiles. In order to analyze the experimental data under the heat conduction theory, we also constructed a 2D axi-symmetric heat transfer solution for a semi-transparent finite membrane using Green's functions. With this approach we obtained two analytical expressions: one for the axial unsteady profile and another for the steady ΔT profile in the r−z plane using the delta function approximation. We found a linear relationship between the experimental ΔT steady values and the optical intensity Io; this linear relationship was maintained until incandescence at the membrane surface appeared. We also found that the steady ΔT values had little dependance on the membrane thickness L; conversely, the transient ΔT values exhibited a clear dependance on L, as expected. The theoretical solution fitted the experimental data only for a thermal diffusivity smaller than that of pristine PDMS. Our analysis suggests that such heat transfer retardation may be due to the nonlinear scattering behavior of the composite or due to additional interfacial resistances imposed by the experiment. Finally, experiments and theory showed that the L−ΔT curves have a maximum point (maximum temperature) at an optimal membrane thickness Lop; this optimal point in turn decreases as the extinction coefficient increases. Additional analysis of the results were also conducted regarding the area of the membranes, and both, experiments and theory, suggest that the heat generation induced by irradiating a surface with optical fibers is indeed highly localized.

A Discrete Fourier Transformation-based Method for Phase Delay Estimation in X-ray Pulsar Navigation

X-ray pulsar navigation is a promising technology for autonomous spacecraft navigation. The key measurement of pulsar navigation is the time delay (phase delay). There are various methods to estimate phase delay, but most of them have high computational complexities. In this paper, a new method for phase delay estimation is proposed, which is based on the time-shift property of Discrete Fourier Transformation (DFT). With this method, the time complexity can be greatly reduced. Also, a delta-function approximation can be used to further decrease the computational cost. Numerical simulation shows that the proposed method is effective for phase delay estimation, and the reduced complexity makes our method more suitable for on board implementation.

A collocation technique for solving nonlinear Stochastic Itô–Volterra integral equations

A numerical method for solving nonlinear Stochastic Itô–Volterra equations is proposed. The method is based on delta function (DF) approximations. The properties of DFs and their operational matrix of integration together with the Newton–Cotes nodes are presented and utilized to reduce the problem to the solution of a nonlinear system of algebraic equations. The method is computationally attractive and some numerical examples are provided to illustrate its high accuracy.

Delta-function Approximation SSC Model in 3C 273

We obtain an approximate analytical solution using δ approximate calculation on the traditional one-zone synchrotron self-Compton (SSC) model. In this model, we describe the electron energy distribution by a broken power-law function with a sharp cut-off, and non-thermal photons are produced by both synchrotron and inverse Compton scattering of synchrotron photons. We calculate the radiation energy spectrum of electrons by the δ function. We apply this model to the multi-wavelength Spectral Energy Distributions (SED) of the 3C 273 in different states, and obtain excellent fits to the observed spectra of this source.

The Observed Linearity and Detection Response of Magnetic Fluid Concentration Magnetometry—A Theoretical and Experimental Description

The response of a scanning column magnetometer (SCM) used to measure concentration profiles of columns of magnetic dispersions has been investigated experimentally and theoretically. From the observed linearity of the total SCM output signal as a function of magnetic particle concentration, a consistent theoretical description is developed that assumes a small sensing coil field and no on-going particle agglomeration. Further theoretical development resulted in a detection coil response function that compared well with the measured response of a delta function approximation in the form of a thin ferrite disk and means that the SCM response function and spatial resolution may be determined from the coil design alone.

On the Dirac Delta Approximation and the MGF Method for ASER Analyses of Digital Communications over Fading Channels

Two distinct methods are applied to derive tighter closed-form approximations for the average symbol error rate (ASER) of a multitude of coherent and differentially coherent digital modulations (with/without diversity) in different fading environments. The first approach relies on the Dirac delta function approximation of a generalized function g(x)=x <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c-1</sup> exp(-ax) to eliminate the need for integration, while the second method exploits a tight exponential-type approximation for the Gaussian probability integral to express the ASER in terms of only the moment generating function (MGF) of received signal-to-noise ratio (SNR) random variable.

A piecewise constant method for Frobenius–Perron operators via delta function approximations

Let S:[0,1]→[0,1] be a nonsingular transformation such that the corresponding Frobenius–Perron operator PS:L1(0,1)→L1(0,1) has a stationary density f∗. We develop a piecewise constant method for the numerical computation of f∗, based on the approximation of Dirac’s delta function via pulse functions. We show that the numerical scheme out of this new approach is exactly the classic Ulam’s method. Numerical results are given for several one dimensional test mappings.

Superposition states of ultracold bosons in rotating rings with a realistic potential barrier

In a recent paper [Phys. Rev. A 82, 063623 (2010)] Hallwood et al. argued that it is feasible to create large superposition states with strongly interacting bosons in rotating rings. Here we investigate in detail how the superposition states in rotating-ring lattices depend on interaction strength and barrier height. With respect to the latter we find a trade-off between large energy gaps and high cat quality. Most importantly, we go beyond the \delta-function approximation for the barrier potential and show that the energy gap decreases exponentially with the number of particles for weak barrier potentials of finite width. These are crucial issues in the design of experiments to realize superposition states.

Periodic Changes Effects of Distance between Magnetic Barriers on Electron Transport in Nanostructure

I theoretically investigate the spin Polarization and transmission of the electrons in a nanostructure consisting barriers with periodic, parallel and also anti parallel magnetization .also I investigate polarization when distance between barriers is constant, or is increased, or is decreased periodically. These observable quantities are found to be strongly affected by both the magnetic configuration and the number of the periodic magnetic barriers. When the number of periods increases, in parallel magnetization for periodic increasing distance the polarization is enhanced so in parallel configuration it is better that distance between barriers to be increasing periodically. I investigate Polarization in these configuration in both delta function approximation and modulated magnetic barriers in x direction. This Polarization can be used in spintronics device.

Delta function approximations in level set methods by distance function extension

In [A.-K. Tornberg, B. Engquist, Numerical approximations of singular source terms in differential equations, J. Comput. Phys. 200 (2004) 462-488], it was shown for simple examples that the then m ...

Controlling Coherence Using the Internal Structure of HardπPulses

The tiny difference between hard pi pulses and their delta-function approximation can be exploited to control coherence. Variants on the magic echo that work despite a large spread in resonance offsets are demonstrated using the zeroth- and first-order average Hamiltonian terms, for 13C NMR in 60C. The 29Si NMR linewidth of silicon has been reduced by a factor of about 70,00 using this approach, which also has potential applications in magnetic resonance microscopy and imaging of solids.

Relaxation During Adiabatic Radiofrequency Pulses

We treat in this review the general problem of relaxation during an adiabatic radiofrequency pulses. In most of the applications literature in Nuclear Magnetic Resonance (NMR) the relaxation during a radio-frequency pulse has either been neglected or avoided using the so-called “delta function approximation” where relaxation during the short time of the pulse is small, when the relaxations during the pulses can be ignored. There has developed within the NMR research community the study and use of pulses on the order of many milliseconds where there is an explicit time-dependence in the first rotating frame (FRF). The result of these developments is that the time-dependence of relaxation during the radiofrequency pulse becomes important for the successful analysis of NMR data. In this review we detail some of the theoretical concepts for treating NMR relaxation that we have found necessary in developing the successful fitting to data of NMR experiments where the time dependence results from the use of adiabatic pulses. We explicitly treat the case of identical spins 1/2, then the situation of two distinct spins 1/2 both for the case of the dipolar relaxation mechanism and for two site chemical exchange.

Investigation of the linear and nonlinear properties of a Drude model photonic crystal

The optical transmission properties of a one-dimensional Drude model photonic crystal are investigated. The linear wave propagation is studied by the transfer matrix method. Using a delta-function approximation a nonlinear 2-d map is obtained and used to explore the global transmission properties of nonlinear wave propagation in this system. Inclusion of a frequency-dependent refractive index represented by the Drude model results in a considerable modification of the characteristics of the linear and nonlinear wave propagation of the one-dimensional photonic crystal.

Asymptotic conditions for the electromagnetic form factors of hadrons represented by the VMD model

A system of linear homogeneous algebraic equations for coupling constant ratios of vector-mesons to hadrons is derived by imposing an asymptotic behaviuor upon the VMD pole parametrization of an hadron electromagnetic form factor. A similar system of equations with a simpler structure of coefficients, taken as even powers of vector-meson masses, is derived by means of integral superconvergent sum rules for the imaginary part of the considered from factor using its appropriate delta function approximation. Although both systems have been derived starting from different properties of the electromagnetic form factor and they look each in its own way, it is shown explicitly that they are fully equivalent.

Simulation of a Flapping Flexible Filament in a Flowing Soap Film by the Immersed Boundary Method

This paper reports the computer simulation of a flapping flexible filament in a flowing soap film using the immersed boundary method. Our mathematical formulation includes filament mass and elasticity, gravity, air resistance, and the two wires that bound the flowing soap film. The incompressible viscous Navier–Stokes equations, which are used to describe the motion of the soap film and filament in our formulation, are discretized on a fixed uniform Eulerian lattice while the filament equations are discretized on a moving Lagrangian array of points which do not necessarily coincide with the fixed Eulerian mesh points of the fluid computation. The interaction between the filament and the soap film is handled by a smoothed approximation to the Dirac delta function. This delta function approximation is used not only to interpolate the fluid velocity and to apply force to the fluid (as is commonly done in immersed boundary computations), but also to handle the mass of the filament, which is represented in our calculation as delta function layer of fluid mass density supported along the immersed filament. Because of this nonuniform density, we need to use a multigrid method for solving the discretized fluid equations. This replaces the FFT-based method that is commonly used in the uniform-density case. Our main results are as follows. (i) The sustained flapping of the filament only occurs when filament mass is included in the formulation of the model; within a certain range of mass, the more the mass of the filament, the bigger the amplitude of the flapping. (ii) When the length of filament is short enough (below some critical length), the filament always approaches its straight (rest) state, in which the filament points downstream; but when the length is larger, the system is bistable, which means that it can settle into either state (rest state or sustained flapping) depending on the initial conditions. This numerical result we observed in computer simulation is the same as that of the laboratory experiment even though the Reynolds number of the computations is lower than that of the laboratory experiment by two orders of magnitude.

Prospects of atomic resolution imaging with an aberration-corrected STEM

We investigated high-resolution scanning transmission electron microscope (STEM) images obtained from a microscope equipped with a spherical aberration corrector. The probe size (full-width at half-maximum) is reduced to 0.76 A at 200 kV by assuming the fifth-order spherical aberration coefficient C5 = 100 mm. For the simulation we have used the recently developed scheme for a STEM image simulation based on the Fast Fourier Transform (FFT) multislice algorithm. The peak-to-background (P/B) ratio of the high-angle annular dark-field (HAADF) image is significantly improved at a thin specimen region. Although the P/B ratio becomes worse at a thicker region, the resolution is kept high even at such a region. An almost true HAADF signal will be obtained even from a weak-scattering phosphorous column in InP [001] when the background is subtracted. In the bright-field image the coherent character of elastic scattering is suppressed by averaging over a large convergence angle, making the specimen effectively self-luminous. The claim that HAADF imaging is relatively insensitive to a defocus as well as a specimen thickness is valid only qualitatively, and a detailed image simulation will be required for a quantitative analysis as in the case of the conventional transmission electron microscope. It was noted that the delta function approximation for the object function may not be applicable for a very fine probe, and that the achievable resolution of the HAADF imaging will be limited by the widths of the high-angle thermal diffuse scattering potential.