Example 2 - Polar Ellipsoidal

A second instructive example is a stretched polar coordinate system (see figure ). A polar ellipsoidal coordinate system is specified by,  

$$\left[\begin{array} {c} x_1\ x_3 \end{array}\right] =... ...3 \ a(\xi_3)\,\xi_1\, \rm{sin}\,\xi_3 \ \end{array}\right].$$ (29)

Parameter $a=a(\xi_3)$ is a smooth function controlling coordinate system ellipticity and has curvature parameters $b=\frac{\partial a}{\partial \xi_3}$ and $c=\frac{\partial^2 a}{\partial \xi_3^2}$.The metric tensor g_ij is,  

$$\left[g_{ij}\right] = \left[\begin{array} {cc} a^2 & \xi_1\, a\, b \ \xi_1 \, a \, b & \xi_1^2(b^2+a^2) \end{array}\right],$$ (30)

with determinant $\left\vert \mathbf{g} \right\vert= a^4 \xi_1^2$. The associated metric and weighted associated metric tensors are given by,  

$$\left[g^{ij}\right] = \left[\begin{array} {cc} \frac{b^2+a^... ...rac{b}{a} \ -\frac{b}{a} & \frac{1}{\xi_1}\end{array}\right].$$ (31)

 

2Dex2 Figure 3 Polar ellipsoidal coordinate system example.

Tensors g^ij and m^ij specify a wavenumber appropriate for extrapolating wavefields on a 2-D non-orthogonal mesh (see equation 41). However, because the coordinate system is spatially variant, we must also compute the n_i fields: $n_1=\frac{a^2+2b^2-ac}{a^2}$ and n₃=0. Inserting these values yields the following extrapolation wavenumber $k_\xi_3$,  

$$k_\xi_3= \frac{\xi_1 b}{a} k_\xi_1\pm \sqrt{ a^2 \xi_1^2 s^2... ...xi_1^2 - i k_\xi_1\xi_1 \left(\frac{a^2+2b^2-ac}{a^2} \right)}.$$ (32)

The kinematic version of equation 32 is,  

$$k_\xi_3= \xi_1 \left[\frac{b}{a} k_\xi_1\pm \sqrt{ a^2 s^2 \omega^2 - k_\xi_1^2 }\right],$$ (33)

while the orthogonal polar case (i.e. a=1) recovers the following,  

$$k_\xi_3= \pm \xi_1 \sqrt{ s^2 \omega^2 -k_\xi_1^2 }.$$ (34)

Figure  shows an ellipsoidal polar coordinate system defined by $a=1+0.2\,\xi_3-0.05\,\xi_3^2$. The upper left panel shows a $v(z)=1500+0.2\,z$ velocity function overlain by a coordinate system mesh. The upper right panel presents velocity model as mapped into the GRWE domain. The data used in this test consisted of 4 flat plane-waves. Given this experimental setup, propagating flat plane-waves should not bend in the Cartesian domain because of the v=v(z) velocity model, even though there is velocity variation across each extrapolation step in the GRWE domain. Hence, the impulses have curvature in the GRWE domain (lower right panel). The lower left panel shows the GRWE domain imaging results mapped back to a Cartesian domain. Consistent with theory, the flat plane-waves are imaged as flat reflectors. Note that the edge effects are again caused by coordinate system and plane-wave truncation.

 

Polar
Figure 4 Ellipsoidal polar coordinate system test example. Upper left: $v(z)=1500+0.2\,z$ velocity function overlain by a polar ellipsoidal coordinate system defined by parameter $a=1+0.2\,\xi_3-0.05\,\xi_3^2$. Upper right: velocity model in the GRWE domain. Bottom right: Imaged reflectors in GRWE domain. Bottom left: the GRWE domain image mapped to a Cartesian mesh.