Regularized Ensemble Kalman Methods For The Acoustic And Elastic Wave Inverse Scattering Problems

The inverse scattering problem is a hot research topic in recent years,which has been applied in various fields such as radar imaging,echo positioning,geophysical imaging,non-destructive testing,etc.The inverse scattering problem is nonlinear and strongly illposed,which means the relationship between the far-field mode and the boundary of the cylinder is non-linear,and small data errors in the far-field mode may cause great changes in the scatterer boundary.Many scholars use the Tikhonov regularization method to eliminate the ill-posedness of the inverse scattering problem by adding prior information.To solve the acoustic and elastic wave inverse scattering problems,this thesis uses statistical methods as the inversion method and adds regularization terms to the ensemble Kalman filter(En KF).The research contents of this thesis are as follows:First,in the acoustic direct scattering problem,this thesis combines single-layer potential and double-layer potential in order to approximate the scattering solution of Helmholtz equation,and then uses the Nystr(?)m method to solve the boundary integral equation to obtain the far-field data of the direct scattering problem.Then,using the obtained far-field data as the known conditions,the regularized En KF is used to invert the barrier boundary,and finally the Metropolis-Hastings(MH)algorithm,the En KF and the regularized En KF are used to invert the barrier boundary.The comparison between the boundary after inversion and the precise boundary proves the effectiveness of the regularized En KF.In the elastic wave inverse scattering problem,the Nystr(?)m method is used to solve the direct problem and obtain the far field data.Then,the regularized En KF is used for inversion.Finally,the inversion boundary is compared with the inversion boundary of En KF and real boundary to verify the effectiveness of the regularized En KF in dealing with elastic wave inverse scattering problem.The inverse scattering problem is nonlinear and ill-posed.In this thesis,En KF is used to transform acoustic inverse scattering and elastic inverse scattering problems into a minimization problem,and regularization term is added to solve the ill-posedness of acoustic inverse scattering and elastic inverse scattering problems.In this thesis,it is proved that the method must have an optimal solution,and the effectiveness of the method is proved by numerical experiments.Aiming at the numerical experiment of acoustic wave inverse scattering problem,three different incident directions are selected with fixed observation directions,and the performance of different algorithms is compared,and the errors between different algorithms are given.Then the incident direction is fixed,three different observation directions are studied,compare algorithms performance and give the errors of different algorithms.In order to solve the problem of elastic wave inverse scattering problem,two different observation directions and three different incident directions are selected to study the P wave case,and the errors of En KF and regularized En KF are compared.In the S wave case,two different observation directions and three different incident directions are also studied,and the reconstruction effects and errors of En KF and regularized En KF are compared.The results show that the method proposed in this thesis is more accurate and of less errors in dealing with inverse scattering problem.