Non-linear seismic wave propagation in anisotropic media using the flux-corrected transport technique

Both anisotropy and non-linearity of rocks are important properties related to the elastic wave propagation in the Earth. However, in the literature, there has been a continuous increase in the interest of anisotropy, but very few studies of non-linearity are reported. In this paper, we aim to tackle this imbalance by examining propagation characteristics of non-linear elastic waves in anisotropic media. As analytic solutions to anisotropic wave equation in non-linear media (or non-linear wave equation in anisotropic media) are not readily available, we use a numerical-based approach. However, we realize that numerical modelling of non-linear seismic waves suffers from problems such as steep gradients, shocks and unphysical oscillations. Accordingly, some special treatments have been presented in the literature to reduce these problems. In this paper, we present a second-order central finite difference scheme based on the modified flux-corrected transport (FCT) technique, and we also present the stability criterion for the FCT. Our modelling results show that the product of strength of non-linearity and the initial amplitudes of the seismic source is the main factor influencing propagation characteristics accumulated with traveltimes. Moreover, the effects of non-linear elastic wave propagation can be distorted by the initial frequency of the seismic source and become enhanced by the presence of anisotropy. Similarly, the effects induced by anisotropy are enhanced in non-linear media. Both in isotropic and vertical symmetry axis media, the non-linear effects, such as the waveform aberration (or waveform distortion), the resonant peak shift and the generation of harmonics, can be clearly seen, and the interaction between anisotropy and non-linearity is also obvious. In conclusion, both the anisotropy and the non-linearity should be considered to investigate characteristics of elastic wave propagation in solid media, especially when the source is strong (e.g. in source regions).