Why Elliptic Coordinates?

Generating a good coordinate system for RWE prestack migration requires appropriately linking mesh geometry with the dynamics of propagating wavefields. Figure  illustrates this for an idealized shot-profile imaging experiment where source and receiver wavefields (S and R) are point sources defined at $[{\bf s},\tau_s=0]$ and $[{\bf r},\tau_r=\tau]$ in a constant velocity medium $v({\bf x})$. In this experimental setup, the wavefields expand outward as spherical wavefronts (dashed lines) described by

$$S({\bf s},{\bf x}; t ) = \delta \left(t - \frac{ \vert\vert{... ...{ \vert\vert{\bf x} - {\bf r}\vert\vert}{v({\bf x})} \right).$$ (1)

An image is generated by applying a correlation imaging condition at t=0,  

$$I ({\bf x}) = \sum_{s} \sum_{r} \delta \left[ \tau - \left( ... ...ert + \vert\vert{\bf x}-{\bf s}\vert\vert}{v} \right) \right],$$ (2)

which is the equation of ellipse (solid line). This suggests a natural link between elliptic coordinate systems and prestack migration, which is illustrated in Figure  by the similarity of the drawn isochron and the coordinate mesh.

 

WhyEC
Figure 1 Idealized imaging experiment in a constant medium. Source and receiver wavefields (dashed lines) are expanding point sources described by fields $S({\bf s},{\bf x}; t )$ and $R( {\bf r}, {\bf x}; t)$. The corresponding image is an elliptic isochron surface $I({\bf x})$ derived by cross-correlating the source and receiver wavefields (solid line). Note that the overlain elliptic coordinate system closely matches (though not identically) the isochron surface suggesting that this is a good coordinate system for RWE prestack shot-profile migration.

The keen observer will note that the foci of the elliptic coordinate system in Figure  were not specified relative to ${\bf s}$ and ${\bf r}$. Shifting these points around alters both the mesh and how well it matches the isochrons. However, this represents two degrees of freedom that allow us to optimally match mesh geometry to the wavefield propagation dynamics.