Example 1 - 2-D Sheared Cartesian Coordinates

An instructive example is a coordinate system formed by a shearing action on a Cartesian mesh (see figure ). A sheared Cartesian coordinate system is defined by,  

$$\left[\begin{array} {c} x_1\ x_2\ x_3 \end{array}\rig... ...egin{array} {c} \xi_1\ \xi_2\ \xi_3\ \end{array}\right],$$ (21)

where $\theta$ is the shear angle of the coordinate system ($\theta=90^{\circ}$ is Cartesian).

 

2Dexamp Figure 1 2-D sheared Cartesian coordinates. Left panel: Physical domain represented by sheared Cartesian coordinates defined by $\{x_1,x_3\}$; Right panel: GRWE domain chosen to be the unit square $\{ \xi_1,\xi_2 \}$.

This system reduces to a more workable set of two equations,  

$$\left[\begin{array} {c} x_1\ x_3 \end{array}\right] =... ... \left[\begin{array} {c} \xi_1\ \xi_3\ \end{array}\right],$$ (22)

that has a metric tensor g_ij given by,  

$$\left[g_{ij}\right] = \left[\begin{array} {cc} \frac{\partia... ... \rm{cos}\, \theta \ \rm{cos}\,\theta & 1 \end{array}\right],$$ (23)

with a determinant $\left\vert \mathbf{g} \right\vert= \rm{sin}^2\,\theta$ and an associated metric tensor g^ij given by,  

$$\left[g^{ij}\right] = \frac{1}{\rm{sin}^2\,\theta} \left[\be... ...- \rm{cos}\,\theta \ -\rm{cos}\,\theta & 1\end{array}\right].$$ (24)

Note that because the tensor in equation 24 is coordinate invariant, equation 10 simplifies to,  

$$\Delta \mathcal{U}= g^{ij} \,\frac{\partial^2 \mathcal{U}}{\partial \xi_i\partial \xi_j} = - \omega^2\ss^2\mathcal{U},$$ (25)

and generates the following dispersion relation,

$$g^{ij}k_\xi_ik_\xi_j= \omega^2\ss^2.$$ (26)

Expanding out these terms leads to an expression for wavenumber $k_\xi_3$,  

$$k_\xi_3= -\frac{ g^{13} }{ g^{33} } k_\xi_1 \pm \sqrt{\frac{... ...33}}-\left( \frac{g^{13}}{g^{33}} \right)^2\right)k_\xi_1^2}.$$ (27)

Substituting the values of the associated metric tensor in equation 24 into equation 27 yields,

$$k_\xi_3= \rm{cos} \, \theta \;k_\xi_1 \pm \rm{sin} \,\theta \sqrt{\ss^2\omega^2 - k_\xi_1^2}.$$ (28)

A numerical test using a Cartesian coordinate system sheared at 25$^{\circ}$ from vertical is shown in figure . The background velocity model is 1500 ms^-1 and the zero-offset data consist of 4 flat plane-waves t=0.2, 0.4, 0.6 and 0.8 s. As expected, the zero-offset migration results image reflectors at depths z=300, 600, 900, and 1200 m. Note that the propagation has created boundary artifacts: those on the left are reflections due to a truncated coordinate system while those on the right are hyperbolic diffractions caused by truncated plane-waves.

 

Rays0 Figure 2 Sheared Cartesian coordinate system test. Coordinate system shear angle and velocity are $\theta=25^{\circ}$ and 1500 ms-1, respectively. Zero-offset data consist of 4 flat plane-wave impulses at t=0.2, 0.4, 0.6 and 0.8 s that are correctly imaged at depths z=300, 600, 900, and 1200 m.