Kirchhoff migration using eikonal-based computation of traveltime source-derivatives

Numerical Implementation

Equation 6 is a linear first-order PDE suitable for upwind numerical methods (Franklin and Harris, 2001). Since it does not change the non-linear nature of the eikonal equation, the resulting traveltime source-derivative can be related to any branch of multi-arrivals, if one supplies the corresponding traveltime in $\hat{T}$. The source-derivatives can be computed either along with traveltimes or separately. In Appendix A, we describe a first-arrival implementation based on a modification of FMM (Sethian, 1996).

The first-order traveltime source-derivative enables a cubic Hermite interpolation (Press et al., 2007). Geometrically, such an interpolation is valid only when the selected wave-front in the interpolation interval is smooth and continuous. For a 2D model and a source interpolation along the inline direction only, the Hermite interpolation reads:

$$\begin{array}{lcl} T (z,x; z_s,x_s + \alpha \Delta x_s) & =... ...ial T}{\partial x_s} (z,x; z_s,x_s + \Delta x_s)\;, \end{array}$$ (10)

where $\alpha \in [0,1]$ controls the source position to be interpolated between known values at $(z_s,x_s)$ and $(z_s,x_s+\Delta x_s)$. For comparison, the linear interpolation can be represented by:

$$T (z,x; z_s,x_s + \alpha \Delta x_s) = (1-\alpha)\,T (z,x; z_s,x_s) + \alpha\,T (z,x; z_s,x_s + \Delta x_s)\;.$$ (11)

The linear interpolation fixes the subsurface image point $(z,x)$. A possible improvement is to instead fix the vector that links the source with the image, such that on the right-hand side the traveltimes are taken at shifted image locations:

$$\begin{array}{lcl} T (z,x; z_s,x_s + \alpha \Delta x_s) & =... ...x + (1-\alpha) \Delta x_s; z_s,x_s + \Delta x_s)\;. \end{array}$$ (12)

We will refer to scheme 12 as shift interpolation. According to our definition of the relative coordinate $\mathbf{q}$ in equation 2, shift interpolation amounts to a linear interpolation in $\hat{T}(\mathbf{q};\mathbf{x_s})$. It is easy to verify that, for a constant-velocity medium, both Hermite and shift interpolations are accurate, while the linear interpolation is not. However, the accuracy of shift interpolation deteriorates with increasing velocity variations, as it assumes that the wave-front remains invariant in the relative coordinate. Equations 10-12 can be generalized to 3D by cascading the inline and crossline interpolations (for example equation 11 in 3D case becomes bilinear interpolation). The interpolated source does not need to lie collinear with source samples.

The derivatives themselves can also be directly used for Kirchhoff anti-aliasing (Lumley et al., 1994; Abma et al., 1999; Fomel, 2002). Equations 10, 11 and 12 give rise to their corresponding source-derivative interpolations after applying the following chain-rule to both sides:

$$\frac{\partial}{\partial (x_s + \alpha \Delta x_s)} = \frac... ...)} = \frac{1}{\Delta x_s} \frac{\partial}{\partial \alpha}\;.$$ (13)

The anti-aliasing application is summarized in Appendix B.

Kirchhoff migration using eikonal-based computation of traveltime source-derivatives