Analytical Solutions of the Elastic Wave Equations for Inclined Layers

Analytical solutions provide accurate approach for wavefield calculations. However, conventional seismic analytical solutions assume simplified boundary conditions and constant layer orientations, which makes them unsuitable for accurately modeling of seismic wave propagation in media with inclined layers. To derive the analytical solutions for the elastic wave equations of inclined layered formations, a natural coordinate system is established. Using coordinate transformation techniques, the boundary conditions and elastic seismic wave equations are transformed into the natural coordinate system. Recursive matrices for seismic wavefields in inclined multi-layered formations are derived in natural coordinate system. The verification results confirm that these derived analytical solutions satisfy both the elastic seismic wave equations and boundary conditions across inclined interfaces. The effects of dipping angles and azimuths on seismic displacements and stresses are studied. The numerical results indicate that seismic wavefields in inclined multi-layered formations exhibit irregular spatial periodicity and complex interference patterns.