Calculation of normal and leaky modes for horizontal stratified models based on a semi-analytical spectral element method

SUMMARY The dispersion curves of surface waves controlled by normal modes have been widely used for the retrieval of subsurface structures. However, only the S-wave velocity structures can be retrieved in most cases because normal modes are primarily sensitive to S-wave velocities. Compared with normal modes, leaky modes, which depend on subsurface structures as well and are more sensitive to P-wave velocities, are rarely applied for subsurface imaging. Besides the difficulties of extracting leaky modes from field data, the calculation of leaky modes also prevents the practical application because traditional methods need to search the leaky-mode roots in the complex frequency-wavenumber domain and thus suffer from root skipping. Recently, some new observation methods support the extraction of multi-order leaky mode dispersion, an effective method for calculating leaky modes is consequently required for further investigation. In this paper, a semi-analytical spectral element method (SASEM) is proposed to solve for the normal and leaky modes of elastic waves propagating in a stratified model with a half-space substrate. The transparent boundary condition and semi-infinite element method are used to model the elastic wavefields in the half-space substrate. Then, a linear eigenvalue problem is derived for the modal calculation. Through simple eigenvalue decomposition, we can obtain the solutions of both normal and leaky modes stably and efficiently without any prior estimations, which makes SASEM very friendly to forward modelling and inversion. Several numerical tests were performed to verify the effectiveness of SASEM, as well as to demonstrate its features of high accuracy and no root skipping. Besides the models composed of several homogeneous layers, SASEM was applied to a vertically inhomogeneous offshore model to demonstrate its wide applicability. Analyses on the oscillations of the solved modes show that the leaky modes differ from the normal modes because of the increasing wavefields in the half-space. Moreover, modal analyses confirm that a part of the leaky modes (guided-P modes) is more dependent on the P waves, whereas the other modes are primarily determined by the S waves. Consequently, the effective calculation of leaky modes makes it possible to constrain the P-wave velocity using leaky-mode dispersion curves.