New Sampling Method For The Interior Inverse Scattering Problem

Inverse acoustic scattering problem is an important research field in the inverse problem of mathematical physics.It has wide application prospects in many practical fields such as geophysical exploration,medical imaging,medical diagnosis,sonar,radar,remote sensing and so on.Typical scattering problems are mostly considered as exterior scattering problems.In this paper,we consider a different class of inverse scattering problems the interior scattering problems.Interior scattering problem is a relatively new research topic,it uses the scattered field information measured inside the bounded cavity to reconstruct the shape of the unknown cavity,and the sources(incident waves)are also inside the cavity.Non-destructive testing in some practical projects,such as the test of the structural integrity of ITER fusion reactor,are usually regarded as this type of problem.The sampling method is a classical method for solving the inverse scattering problem.It has the advantage of“nearly no requirement of a priori knowledge for the unknown scatterer",but the disadvantage of the traditional sampling method is that it needs to measure multiple data of the scattering field,which is often not easy to achieve in practical problems.In addition,the amount of calculation is large.In view of this point,this paper studies and analyzes a new sampling method for interior inverse acoustic scattering problems based on the previous work,and tries to reconstruct the shape of the unknown cavity using the single-point source data.When constructing the new sampling method for interior inverse scattering problem,we need to calculate the scattering field generated by a given bounded region,that is,to solve the interior direct scattering problem.Therefore,in Chapter 3,the classical Nystrom method is introduced into the interior direct scattering problem.And the analytical expression of the interior direct scattering problem in a circular region is derived.Finally,numerical examples are given for three different regions.What's more,for the circular region,we will also calculate the analytical results of its scattering field in order to compare with the numerical results of the Nystrom method,so as to verify the effectiveness of the numerical method in this paper.In Chapter 4,the specific theoretical analysis of the new sampling method is given.The scattering field generated by a known bounded region is taken as the core of integral operator.For a given single point source,the measured scattering field data generated by the unknown cavity is moved to right hand sides of the equation.Then the near-field equation of this new sampling method is established.The shape of the unknown cavity is reconstructed by the different properties of the solution of the near-field equation.The new sampling method can process single point source data well,avoid the measurement of multiple data.