468 research outputs found
Eigenvector Model Descriptors for Solving an Inverse Problem of Helmholtz Equation: Extended Materials
We study the seismic inverse problem for the recovery of subsurface
properties in acoustic media. In order to reduce the ill-posedness of the
problem, the heterogeneous wave speed parameter to be recovered is represented
using a limited number of coefficients associated with a basis of eigenvectors
of a diffusion equation, following the regularization by discretization
approach. We compare several choices for the diffusion coefficient in the
partial differential equations, which are extracted from the field of image
processing. We first investigate their efficiency for image decomposition
(accuracy of the representation with respect to the number of variables and
denoising). Next, we implement the method in the quantitative reconstruction
procedure for seismic imaging, following the Full Waveform Inversion method,
where the difficulty resides in that the basis is defined from an initial model
where none of the actual structures is known. In particular, we demonstrate
that the method is efficient for the challenging reconstruction of media with
salt-domes. We employ the method in two and three-dimensional experiments and
show that the eigenvector representation compensates for the lack of low
frequency information, it eventually serves us to extract guidelines for the
implementation of the method.Comment: 45 pages, 37 figure
True amplitude one-way propagation in heterogeneous media
This paper deals with the numerical analysis of two one-way systems derived
from the general complete modeling proposed by M.V. De Hoop. The main goal of
this work is to compare two different formulations in which a correcting term
allows to improve the amplitude of the numerical solution. It comes out that
even if the two systems are equivalent from a theoretical point of view,
nothing of the kind is as far as the numerical simulation is concerned. Herein
a numerical analysis is performed to show that as long as the propagation
medium is smooth, both the models are equivalent but it is no more the case
when the medium is associated to a quite strongly discontinuous velocity
Signal and noise in helioseismic holography
Helioseismic holography is an imaging technique used to study heterogeneities
and flows in the solar interior from observations of solar oscillations at the
surface. Holograms contain noise due to the stochastic nature of solar
oscillations. We provide a theoretical framework for modeling signal and noise
in Porter-Bojarski helioseismic holography. The wave equation may be recast
into a Helmholtz-like equation, so as to connect with the acoustics literature
and define the holography Green's function in a meaningful way. Sources of wave
excitation are assumed to be stationary, horizontally homogeneous, and
spatially uncorrelated. Using the first Born approximation we calculate
holograms in the presence of perturbations in sound-speed, density, flows, and
source covariance, as well as the noise level as a function of position. This
work is a direct extension of the methods used in time-distance helioseismology
to model signal and noise. To illustrate the theory, we compute the hologram
intensity numerically for a buried sound-speed perturbation at different depths
in the solar interior. The reference Green's function is obtained for a
spherically-symmetric solar model using a finite-element solver in the
frequency domain. Below the pupil area on the surface, we find that the spatial
resolution of the hologram intensity is very close to half the local
wavelength. For a sound-speed perturbation of size comparable to the local
spatial resolution, the signal-to-noise ratio is approximately constant with
depth. Averaging the hologram intensity over a number of frequencies above
3 mHz increases the signal-to-noise ratio by a factor nearly equal to the
square root of . This may not be the case at lower frequencies, where large
variations in the holographic signal are due to the individual contributions of
the long-lived modes of oscillation.Comment: Submitted to Astronomy and Astrophysic
Automatically Adapted Perfectly Matched Layers for Problems with High Contrast Materials Properties
AbstractFor the simulation of wave propagation problems, it is necessary to truncate the computational domain. Perfectly Matched Layers are often employed for that purpose, especially in high contrast layered materials where absorbing boundary conditions are difficult to design. In here, we define a Perfectly Matched Layer that automatically adjusts its parameters without any user interaction. The user only has to indicate the desired decay in the surrounding layer. With this Perfectly Matched Layer, we show that even in the most complex scenarios where the material contrast properties are as high as sixteen orders of magnitude, we do not introduce numerical reflections when truncating the domain, thus, obtaining accurate solutions
A Secondary Field Based hp-Finite Element Method for the Simulation of Magnetotelluric Measurements
In some geophysical problems, it is sometimes possible to divide the subsurface resistivity distribution as a one dimensional (1D) contribution plus some two dimensional (2D) inhomogeneities. Assuming this scenario, we split the electromagnetic fields into their primary and secondary components, the former corresponding to the 1D contribution, and the latter to the 2D inhomogeneities. While the primary field is solved via an analytical solution, for the secondary field we employ a multi-goal oriented self-adaptive hp-Finite Element Method (FEM). To truncate the computational domain, we design a Perfectly Matched Layer (PML) that automatically adapts to high-contrast materials that appear in the subsurface and in the air–ground interface. Numerical results illustrate the robustness of the proposed PML and the gains of the secondary field approach, where we obtain results with comparable accuracy than with a full field based formulation but with a much lower computational cost
L²(H¹γ) finite element convergence for degenerate isotropic Hamilton–Jacobi–Bellman equations
In this paper we study the convergence of monotone P1 finite element methods for fully nonlinear Hamilton–Jacobi–Bellman equations with degenerate, isotropic diffusions. The main result is strong convergence of the numerical solutions in a weighted Sobolev space L²(H¹γ(Ω)) to the viscosity solution without assuming uniform parabolicity of the HJB operator
Derivation of high order absorbing boundary conditions for the Helmholtz equation in 2D
We present high order absorbing boundary conditions (ABC)for the Helmholtz equation in 2D, that can adapt to any regular shapedsurfaces. The new ABCs are derived by using the technique ofmicro-diagonalisation to approximate the Dirichlet-to-Neumann map.Numerical results on different shapes illustrate the behavior of thenew ABCs along with high-order finite elements
High-frequency analysis of the efficiency of a local approximate DtN2 boundary condition for prolate spheroidal-shaped boundaries
The performance of the second-order local approximate DtN boundary condition suggested in [4] is investigated analytically when employed for solving high-frequency exterior Helmholtz problems with elongated scatterers. This study is performed using a domain-based formulation and assuming the scatterer and the exterior artificial boundary to be prolate spheroid. The analysis proves that, in the high-frequency regime, the reflected waves at the artificial boundary decay faster than 1/(ka)15/8, where k is the wavenumber and a is the semi-major axis of this boundary. Numerical results are presented to illustrate the accuracy and the efficiency of the proposed absorbing boundary condition, and to provide guidelines for satisfactory performance
Exponential decay of high-order spurious prolate spheroidal modes induced by a local approximate dtn exterior boundary condition
We investigate analytically the asymptotic behavior of high-order spurious prolate spheroidal modes induced by a second-order local approximate DtN absorbing boundary condition (DtN2) when employed for solving high-frequency acoustic scattering problems. We prove that these reflected modes decay exponentially in the high frequency regime. This theoretical result demonstrates the great potential of the considered absorbing boundary condition for solving efficiently exterior high-frequency Helmholtz problems. In addition, this exponential decay proves the superiority of DtN2 over the widely used Bayliss-Gunsburger-Turkel absorbing boundary condition
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