The Alekseev–Mikhailenko method applied to P–SV wave propagation in an elastic medium

The Alekseev–Mikhailenko method (AMM) is the name given to a series of algorithms that use one or more finite spatial transforms to reduce the dimensionality of a wave-propagation problem to that of one space dimension and time. This reduced equation is then solved using finite-difference techniques, and the space–time solution is recovered by applying inverse finite spatial transform(s). In this paper the elastodynamic wave equation that governs the coupled P–Sv motion in an isotropic, vertically inhomogeneous elastic half space is investigated using the AMM. Two types of impulsive body forces that may be used to excite the medium are examined, as is the problem of obtaining accurate transformed finite-difference analogues at the free surface. The second of these is accomplished by introducing the boundary conditions that the shear and normal stress must vanish here and by incorporating their transforms into the transformed elastodynamic equations. The stability criterion for the explicit finite-difference method is given cursory treatment, as detailed discussion of this aspect may be found in many texts that deal with the subject of finite differences.A coal-seam model (two thin, low-velocity layers embedded in a half space) illustrates the method. Both horizontal and vertical seismic traces are computed for this model and the results examined in relation to other seismic-modelling techniques.