When a thin periodic layer meets corners: asymptotic analysis of a singular Poisson problem

The present work deals with the resolution of the Poisson equation in a bounded domain made of a thin and periodic layer of finite length placed into a homogeneous medium. We provide and justify a high order asymptotic expansion which takes into account the boundary layer effect occurring in the vicinity of the periodic layer as well as the corner singularities appearing in the neighborhood of the extremities of the layer. Our approach combines the method of matched asymptotic expansions and the method of periodic surface homogenization, and a complete justification is included in the paper or its appendix.Comment: 58 page

Signal to Noise Ratio estimation in passive correlation-based imaging

We consider imaging with passive arrays of sensors using as illumination ambient noise sources. The first step for imaging under such circumstances is the computation of the cross correlations of the recorded signals, which have attracted a lot of attention recently because of their numerous applications in seismic imaging, volcano monitoring, and petroleum prospecting. Here, we use these cross correlations for imaging reflectors with travel-time migration. While the resolution of the image obtained this way has been studied in detail, an analysis of the signal-to-noise ratio (SNR) is presented in this paper along with numerical simulations that support the theoretical results. It is shown that the SNR of the image inherits the SNR of the computed cross correlations and therefore it is proportional to the square root of the bandwidth of the noise sources times the recording time. Moreover, the SNR of the image is proportional to the array size. This means that the image can be stabilized by increasing the size of the array when the recorded signals are not of long duration, which is important in applications such as non-destructive testing

Signal to noise ratio analysis in virtual source array imaging

We consider correlation-based imaging of a reflector located on one side of a passive array where the medium is homogeneous. On the other side of the array the illumination by remote impulsive sources goes through a strongly scattering medium. It has been shown in [J. Garnier and G. Papanicolaou, Inverse Problems 28 (2012), 075002] that migrating the cross correlations of the passive array gives an image whose resolution is as good as if the array was active and the array response matrix was that of a homogeneous medium. In this paper we study the signal to noise ratio of the image as a function of statistical properties of the strongly scattering medium, the signal bandwidth and the source and passive receiver array characteristics. Using a Kronecker model for the strongly scattering medium we show that image resolution is as expected and that the signal to noise ratio can be computed in an essentially explicit way. We show with direct numerical simulations using full wave propagation solvers in random media that the theoretical predictions based on the Kronecker model are accurate

Numerical resolution of the wave equation on a network of slots

In this technical report, we present a theoretical and numerical model to simulate wave propagation in finite networks of rods with both classical Kirchhoff conditions and Improved Kirchhoff conditions at the nodes of the networks. One starts with the continuous framework, then we discretize the problem using finite elements with the mass lumping technic introduced by G.~Cohen and P.~Joly. Finally, we show an implementation of the obtained numeric scheme in a homemade code written in C++ in collaboration with K.~Boxberger, some results and some error estimates

Propagation of an acoustic wave in a junction of two thin slots

In this research report, we analyze via the theory of matched asymptotics the propagation of a time harmonic acoustic wave in a junction of two thin slots. This allows us to propose improved Kirchoff conditions for the 1D limit model. These conditions are analyzed and validated numerically

Study of propagation of acoustic waves in junction of thin slots

In this document, we analyze via the theory of matched asymptotics the propagation of a time domain acoustic wave in a junction of thin slots. This allows us to propose Improved Kirchhoff conditions for the 1D limit problem. These conditions are analyzed and validated numerically

On the homogenization of microstructured surfaces

The direct numerical simulation of microstructured interfaces like multiperforated absorber in acoustics with hundreds or thousands of tiny openings would result in a huge number of basis functions to resolve the microstructure. One is, however, primarily interested in the effective and so homogenized transmission and absorption properties. We introduce the surface homogenization that asymptotically decomposes the solution in a macroscopic part, a boundary layer corrector close to the interface and a near field part close to its ends. The introduction is for a general framework of models of elliptic partial differential equations incorporating the influence of end-points of the microstructured interfaces to the macroscopic part of the solution. The effective transmission and absorption properties are expressed by transmission conditions on an infinitely thin interface and corner conditions at its end-points to ensure the correct singular behaviour, intrinsic to the microstructure. We give details on the computation of the effective parameters and show their dependence on geometrical properties of the microstructure on the example of the wave propagation described by the Helmholtz equation. Numerical experiments indicate with the obtained macroscopic solution representation one can reach very high accuracies with a small number of basis functions

SIGNAL TO NOISE RATIO ESTIMATION IN PASSIVE CORRELATION BASED IMAGING

ABSTRACT We consider here the problem of imaging using passive incoherent recordings due to ambient noise sources. The first step towards imaging in this configuration is the computation of the cross-correlations of the recorded signals. These cross-correlations are computed between pairs of sensors (receivers) and they contain very important information about the background medium. Indeed, it was shown both experimentally [1] and theoretically [2] that the Green's function between two sensors can be retrieved from the crosscorrelation of passive incoherent recordings at these sensors. Here, we propose to employ these cross-correlations for imaging reflectors using a travel time migration method. The signal to noise ratio analysis of the proposed method is carried out