Elliptic Coordinate Extrapolation

Propagating wavefields on elliptic meshes using RWE requires incorporating the geometry of the coordinate system directly in the extrapolation equations. This section derives the equations for propagation in the elliptic direction using the non-orthogonal RWE theory developed in Shragge (2006).

The analytic transformation between elliptic and Cartesian coordinate systems (see example in figure ) is specified by,  

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

where [x₁,x₃] are the underlying Cartesian coordinate variables, $[\xi_1,\xi_3]$ are the RWE elliptic coordinates, and a is a stretch parameter controlling the breadth of the coordinate system.

The metric tensor ($g_{ij} = \frac{\partial x_k}{\partial \xi_i}\frac{\partial x_k}{\partial \xi_j}$ with an implicit sum over index k) describing the geometry of the elliptic coordinate system is given by,  

$$\left[g_{ij}\right] = \left[\begin{array} {cc} A^2 & 0\\ 0 & A^2 \\ \end{array}\right],$$ (4)

where $A=a \sqrt{ {\rm sinh}^2 \,\xi_3 + {\rm sin}^2 \,\xi_1 }$. The determinant of the metric tensor is $\left\vert \mathbf{g} \right\vert= A^4$, leading to an associated (inverse) metric tensor given by,  

$$\left[g^{ij}\right] = \left[\begin{array} {cc} A^{-2} & 0 \\ 0 & A^{-2} \\ \end{array}\right].$$ (5)

The weighted metric tensor (the product of the metric tensor and the determinant: $m^{ij}=\sqrt{\left\vert \mathbf{g} \right\vert}\, g^{ij}$) is given by,  

$$\left[m^{ij}\right] = \left[\begin{array} {ccc} 1 & 0 \\ 0 & 1 \\ \end{array}\right].$$ (6)

The corresponding extrapolation wavenumber is generated by using tensors g^ij and m^ij in the wavenumber expression for general 3D non-orthogonal coordinate systems (equations 13 and 14 in Shragge (2006)). Note that even though the elliptic coordinate system varies spatially, the local curvature parameters in the weighted metric tensor are constant. Thus, the extrapolation wavenumber, $k_{\xi_3}$, for recursive wavefield extrapolation stepping outward in concentric ellipses is,  

$$k_{\xi_3} = \pm \sqrt{ A^2 s^2 \omega^2 - k_{\xi_1}^2 },$$ (7)

where $A=a \sqrt{ {\rm sinh}^2 \,\xi_3 + {\rm sin}^2 \,\xi_1 }$, s is slowness (inverse of velocity), and $k_{\xi_1}$ is the orthogonal wavenumber. Equation 7 is an exact representation of the extrapolation wavenumber and does not contain any kinematic approximations. The most striking observation about this expression is that the sole difference between propagation in elliptic and Cartesian coordinate systems is a smooth multiplicative slowness model stretch. Otherwise, existing Cartesian extrapolators can be used for propagating wavefields.