An eikonal based formulation for traveltime perturbation with respect to the source location

Discussion

Seismic data are usually acquired with geophone layouts that record information from multiple source locations. The redundancy in the coverage is necessary to eliminate gaps in the data, estimate velocity, and image the data. Thus, the direct relation between the traveltime field and the source location allows us to estimate attributes that can help in interpolation, velocity estimation and possibly imaging. Specifically:

• Traveltime compression schemes (Alkhalifah, 2010) require the ability to interpolate subsampled traveltimes.

• One of the main sources of trace interpolation information is the behavior of wavefronts with respect to the source location.
• Velocity estimation relies directly on the change in traces as a function of source and receiver locations.

• Kirchhoff antialiasing schemes (Abma et al., 1999; Lumley et al., 1994) explicitly require the derivatives of the traveltime with respect to the source and receiver locations on the surface.

• Gaussian-beam migration (Gray, 2005; Hill, 1990,2001; Alkhalifah, 1995) relies on traveltime derivative information as a function of ray angle variations and source variations, which is usually obtained from the slower and less stable dynamic ray tracing.
• Efficient traveltime calculation can be achieved using first order linear equations instead of the nonlinear form of the eikonal equation.

However, the formulation developed here has limitations. Chief among them is the need to evaluate derivatives of velocity and traveltime fields. Since velocities may include discontinuities, their derivatives are not easy to evaluate. However, similar to the ray-based methods, one can simply smooth the velocity field.

Since traveltime fields that contain all arrivals satisfy the eikonal equation, the source perturbation can be applied to traveltimes extracted from other methods, such as ray tracing or escape equations. Therefore, they can include most energetic arrivals as opposed to first-arrival traveltimes.

An eikonal based formulation for traveltime perturbation with respect to the source location