Anisotropic information regarding the subsurface is important for various geophysical applications, including fracture characterization and estimation of the in situ stress field. Conventional techniques used to invert seismic amplitudes for anisotropic parameters estimate quantities such as the anisotropic gradient, which is not easily interpretable due to the amalgamation of various elastic stiffness coefficients. To gain an improved understanding of the anisotropic parameters estimated from azimuthal amplitude variation with offset (AVO) inversion, we simplify the PP reflection coefficient for a medium with arbitrary anisotropy by imposing an elliptic orthorhombic constraint. This results in a reduced form of the reflection coefficient in which the azimuthally dependent components are only a function of the horizontal P- and vertical S-wave velocities in a rotated coordinate frame. Furthermore, the anisotropic components are parameterized as direct perturbations from the isotropic solution, which requires no a priori information regarding the anisotropy and stabilizes the inversion. We demonstrate the utility of the elliptic orthorhombic approximation through an example in the Montney tight-gas play in northeast British Columbia, Canada, where we perform an azimuthal AVO inversion and estimate the crack density and differential stress ratio.