Numerical analysis of the acoustic wave equation for orthorhombic anisotropy
Orthorhombic anisotropy allows us to add azimiuthul variations, caused primarely by vertical fractures, to the widely recognized and used transversely isotropic with vertical symmetry (VTI) model. In this case, an additional three parameters are needed to describe the medium in the acousitc approximation. As a result, the acoustic orthorhombic anisotropic wave equation is a sixth-order linear partial differential equation with three sets of possible solutions corresponding to the ingoing and outgoing $P$-wave, as well as four other solutons steming from the two-shear wave components, both ingoing and outgoing. Analytical analysis of the solutions of this new equation in a homogeneous medium show that stability of the solutions occur
only when
Similar to the transversely isotropic case, these additional solutions can be lation derived under the acoustic medium
assumption for $P$-waves in orthorhombic anisotropic media, I obtain an acoustic wave equation
valid under the same assumption. Although this assumption is physically impossible for anisotropic media, it
results in wave equations that are kinematically and dynamically accurate for elastic media.
The orthorhombic
acoustic wave equation, unlike the transversely isotropic (TI) one, is a six-order equation with
three sets of complex conjugate solutions. Only one set of these solutions are perturbations
of the familiar acoustic wavefield solution
in isotropic media for in-coming and out-going $P$-waves, and thus, are of interest here. The other two sets of solutions
are simplify the result of this artificially derived
sixth order equation, and thus, represent unwanted artifacts. Like in the TI case, these artifacts
can be eliminated by placing the source in an isotropic layer, where such artifacts do not exist.