Time Reversal Method For Indentifying Multiple Sources

In order to meet the energy needs of modern society,mining work is necessary.In the mine operation,the chain collapse accident of coal mine sometimes happens,which is caused by the vibration of multiple seismic sources at different times.In this dissertation,we use the wave equation to describe the chain collapse phenomenon and explore the source and the time of vibration.This kind of problem belongs to the inverse source problems.There are two kinds of classical inversion methods for the inverse source problem in time domain: one is the time reversal mirror method and the other is the source time reversal method.The above two methods are only applicable to the homogeneous wave equation,but most of this vibration phenomena are usually described by the inhomogeneous wave equation in practical engineering applications.Therefore,in this dissertation,we propose a novel inversion method for reconstructing the source function of the inhomogeneous wave equation.In this dissertation,we study the source inversion of the acoustic and elastic wave cases,respectively.Recently,a lot of works are concerned with the inversion of the spatial term with the time known since the inversion of the source function is very complex when the time term is unknown.In this dissertation,we mainly consider the reconstruction of the time term and spatial term for the wave equation.For the inverse source problem of the acoustic wave case,based on the time reversal method,we firstly reconstruct the spatial term of the source function from the measurement data obtained from the boundary detector.Secondly,we determine the time term by using the Fourier transform.In addition,in order to verify the feasibility of the method,we also show the uniqueness and stability of the reconstructed method.The theoretical results are presented to verify that the inversion method is effective to reconstruct the source function of seismic sources.However,in practical applications,elastic wave model could simulate the chain collapse process better.Therefore,we further explore inverse source problem of reconstruction elastic source.For this problem,we firstly decompose the elastic wave into P wave and S wave according to the Helmholtz decomposition.Then,according to the time reversal method,we can identify the spatial term of the source term by using the boundary measurements.In addition,we also establish the uniqueness and stability for the reconstruction of spatial term.In the stability analysis,P wave and S wave are respectively analyzed to reduce the complexity of the analysis process.Finally,we obtain the time term of the source function by using the Fourier transform and the reconstructed result of the space term.