The pioneer study of simulating the wave field in media with irregular interface belongs to Aki and Lamer. 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.
Head waves are usually considered to be the refracted waves propagating along flat interfaces with an underlying higher velocity.However,the path that the rays travel along in media with irregular interfaces is not clear.Here we study the problem by simulation using a new approach of the spectral-element method with some overlapped elements（SEMO） that can accurately evaluate waves traveling along an irregular interface.Consequently,the head waves are separated from interface waves by a time window.Thus,their energy and arrival time changes can be analyzed independently.These analyses demonstrate that,contrary to the case for head waves propagating along a flat interface,there are two mechanisms for head waves traveling along an irregular interface：a refraction mechanism and transmission mechanism.That is,the head waves may be refracted waves propagating along the interface or transmitted waves induced by the waves propagating in the higher-velocity media.Such knowledge will be helpful in constructing a more accurate inversion method,such as head wave travel-time tomography,and in obtaining a more accurate model of subsurface structure which is very important for understanding the formation mechanism of some special areas,such as the Tibetan Plateau.
