Source time reversal methods for acoustic and elastic waves

Abstract

This dissertation studies the detection and reconstruction of the space term for a variable separable source in acoustic and elastic wave problems. To do this, we study the time-reversal mirror method, which exploits an intrinsic invariance of physics at microscopic level that is also observed at macroscopic level for wave equations. This means that it is possible to recover the initial condition of a homogeneous wave equation by reversing the wave through time. To locate and reconstruct the space term of the source, we develop a method called source time reversal. The underlying application here is the seismic source detection in mining. It is known that mining activity induces tremors inside mines. This become very dangerous if the adequate precautions are not considered. Knowing the origin of the seismic activities could be used to reduce the collapse hazard and improve the safety inside mines. This work is divided into three chapters; each of them constitutes a self-contained document to be presented as an article. The first chapter deals with the problem of source reconstruction for acoustic waves. To do this, we introduce the source time reversal method, which reconstructs the space term of a source of the form f(x)g(t), where f(x) gives the shape and g(t) represents the time distribution of the source. We also present an error estimate for the reconstruction for the case when f is a square-integrable function. Here, we propose a regularization method to implement the source reconstruction numerically. Additionally, we analyze numerically the main features and limitations of the proposed method when applied to acoustic waves. Chapter two studies the problem of source reconstruction for elastic waves. We thus extend the source time reversal method to elastic problems. We also introduce a new regularization method to implement the space-source term reconstruction numerically for a large datasets. The new regularization method eliminates the high frequencies present in the processed signals, which allows coarser numerical meshes and reduces the computational cost. Finally, this chapter presents several numerical experiments to prove that the method is valid in the elastic case. The last chapter analyzes different source reconstruction problem. Here we consider a source composed by a finite sum of variable separable functions, where each time-source term is a Dirac delta function acting at a different time. Based on a time reversal property, the source can be located by observing the displacement and the displacement velocity of the reversed problem. We extend this idea to systems of elastic waves. Additionally, we propose an algorithm for its numerical implementation.