Theory And Application Of Acoustic Inverse Scattering Based On The Nonlinear Weighted Total Variationr

Inverse scattering problem is a typical mathematical physics problem,mainly including acoustic wave,electromagnetic wave,elastic wave inverse scattering.They are widely used in many practical fields such as biology,medical imaging,geology,remote sensing sonar and so on.The scattering problem determines the scattering field according to the incident field and the known information of the inhomogeneous medium,which is mainly solved by artificial boundary conditions and the Lippmann-Schwinger integral equation.The principle is that the scattering phenomenon occurs when the incident wave hits the obstacle body,and the far-field u_∞（or near-field u_s）data of the scattered field can be measured by the sensor.The inverse scattering problem consists of retrieving the location,size,shape and physical properties of an obstacle from sensor measurements.Exploring the physical properties is also called the inverse media problem.The acoustic inverse dielectric scattering problem is nonlinear and ill-posed.In this thesis,we prove that the near-field data F is Fr（?）chet differentiable,and（F[q]）^-1is also Fr（?）chet differentiable.We then use the Fr（?）chet derivative to linearize the complex acoustic inverse scattering problem.For its ill-posedness,the regularization method is generally used to narrow the solution space to overcome its ill-posedness.However,the traditional Tikhonov regularization method is too smooth,resulting in blurring of image edges.The traditional Total Variation（TV）regularization method will produce staircase effect in the image inversion.In recent years,the regularization method of joint sparse and TV is time-consuming,and theL₁/L₂regularization method does not have region dependence,so it is difficult to remove the distortion along the coordinate axis.To solve the above problems,the anisotropic TV regularization method is ap-plied to the acoustic inverse dielectric scattering,and several two-dimensional nu-merical experiments are done to verify the results.It is found that the anisotropic TV regularization will distort along the number line.Therefore,in this thesis,the nonlinear weighted anisotropic total variation（NWATV）regularization method is used to avoid the distortion along the coordinate axis.In order to verify the ef-fectiveness of this method,2D and 3D numerical experiments are carried out.The results show that the NWATV regularization method can well preserve the internal non-uniform structure compared with the regularization of the joint sparsity and TV and the recently proposed L₁/L₂regularization method.