Modeling of pseudo-acoustic P-waves in orthorhombic media with a lowrank approximation

Tilted Orthorhombic Anisotropy

Tectonic movement of the crust may rotate the rocks and tilt the plane containing the vertical cracks, causing a tilted anisotropy. In the case of tilted orthorhombic media, $k_x$, $k_y$, and $k_z$ need to be replaced by $\hat{k}_x$, $\hat{k}_y$, and $\hat{k}_z$, which are spatial wavenumbers evaluated in a rotated coordinate system aligned with the vectors normal to the orthorhombic symmetry planes:

$$\begin{array}{*{20}c} \hat{k}_x=k_x\cos{\phi}+k_y\sin{\phi}\... ...a}-k_y\cos{\phi}\sin{\theta}+k_z\cos{\theta}\;,\\ \end{array}$$ (14)

where $\theta$ is the dip angle measured with respect to vertical and $\phi$ is the azimuth angle, which is the angle between the original X-coordinate and the rotated one. The original vertical axis has the direction of $\left\{\sin\theta\sin\phi,-\sin\theta\cos\phi,\cos\theta\right\}$. For a more general rotation, one needs three angles to describe the transformation (Zhang and Zhang, 2011).