Wave-equation generated Green's Functions

Generating high-quality source Green's function estimates is an important component of seismic imaging. Because wave-equation methods naturally handle wavefield triplication, Green's functions developed using corresponding operators should normally be superior to those generated by other methods. However, implementing the approximations required to handle propagation through laterally variant velocity profiles often can lead to inferior Green's function estimates. These errors become increasingly more apparent, both kinematically and dynamically, as wavefield propagation directions become increasingly oblique to the extrapolation axis.

One solution is to pose wavefield extrapolation in a ray-based coordinate system defined by an extrapolation axis oriented along the axis of increasing travel time and additional coordinates represented by shooting angles Sava and Fomel (2005). However, grids thus specified exhibit attributes that depend intrinsically on the chosen ray-tracing method. For example, ray-coordinate systems generated by Huygen's ray-tracing Sava and Fomel (2001) may triplicate and cause numerical instability during wavefield extrapolation. Hence, care must be taken to ensure that ray-coordinate systems have the appropriate attributes.

One method for calculating singular-valued travel times is with a fast-marching Eikonal equation solver, which provides a travel-time map to each subsurface model location for a given shot point. A travel-time map example is shown in the upper left panel of figure  for a velocity slice of the SEG-EAGE salt data set.

 

Shotpoint
Figure 2 Wave-equation generated Green's functions example. Top left: Travel-time map generated by fast marching eikonal (FMEikonal) equation solver for a slice of the SEG-EAGE salt model overlain by 0.2 s time contours. Top right: A blended coordinate system developed using time contours as the interior and exterior surfaces. This allows for the the coordinate system to be fairly conformal with the orientation of the propagating wavefield. Bottom right: Coordinate mesh output from the differential mesh algorithm after 20 iterations. Note that the kinks in the model have been significantly reduced. Bottom left: Underlying velocity model mapped into the ray-coordinate shown in bottom right.

A ray-based coordinate system can be formed by choosing two isochrons that represent the initial and maximal extrapolation times. The coordinate system is fully defined by connecting the two isochrons with the extremal rays. Blending functions can then be used to specify an intermediate geometry $\{ s^1,s^2 \}$. (see upper right panel of figure ).

Coordinate systems generated with this approach, though, are not guaranteed to be smooth and generally will require mesh regularization. The bottom right panel shows the output of the differential grid generation algorithm $\{x^1,x^2\}$ after 20 smoothing iterations. Note that kinks visible in the upper right panel have disappeared leaving a significantly smoother mesh. A qualitatively test of coordinate system smoothness is to examine the smoothness of the underlying velocity model in the transform domain. The salt body example (lower left panel) indicates that the velocity model is fairly smooth and should not create significant problems for generalized coordinate wavefield extrapolation.