Review on localized boundary integral equation: Discrete wavenumber method for 2D irregular layers

The pioneer study of simulating the wave field in media with irregular interface belongs to Aki and Larner. Since that many numerical methods on the subject have been developed, such as pure numerical techniques, ray method and boundary method. The boundary method based on boundary integral equation is a semi-analytical method which is suitable to modeling wave field induced by irregular border. According to the property of the applied Green's function the boundary methods can be sorted into space domain boundary method and wavenumber domain boundary method. For both of them it is necessary to solve a large equation, which means much computation is needed. Thus, it is difficult for the boundary methods to be applied in simulating wave field with high frequency or in large range. To develop a new method with less computation is meaningful. For this purpose, localized boundary integral equation, i.e., discrete wavenumber method is proposed. It is rooted in the Bouchon-Campillo method, an important wavenumber domain boundary method. Firstly the force on interface is separated into two parts: one is on flat part and the other on irregular part of the interface. Then Fourier transform is applied to identify their relation, the unknown distributes only on irregular part. Consequently computation efficiency is dramatically improved. Importantly its accuracy is the same as that of Bouchon-Campillo.