Numerical implementation

Using the proposed recursion 12 and the layer traveltime derivatives in equation 15, we can summarize the steps to accumulate the effects from lateral heterogeneity along a raypath and evaluate the corresponding traveltime derivatives at the surface as follows:

1. Given a multi-layered medium with known $F$, $W$, and their derivatives in all sublayers, we compute the layer traveltime derivatives (equation 15) for a specified CMP location in the case of reflection traveltime or a specified image-ray escape location in the case of diffraction traveltime. In particular, equation 15 can be rewritten with evaluated reference 1-D traveltime derivatives as follows:

   $$\left.\frac{\partial^2 t_{k-1}}{\partial x_{k-1} \partial x_{k}}\right\rvert_{h=0} = -\frac{1}{T_{k-1} V^2_{nmo,k-1}} + _1 ~,$$ (18)

   $$\left.\frac{\partial^2 t_{k-1}}{\partial x_{k}^2}\right\rvert_{h=0} = \frac{1}{T_{k-1} V^2_{nmo,k-1}} + _2 ~,$$

   $$\left.\frac{\partial^2 t_{k}}{\partial x^2_{k}}\right\rvert_{h=0} = \frac{1}{T_{k} V^2_{nmo,k}} + _3 ~,$$

   where $V_{nmo,k}$ is the NMO velocity of the $k$-th layer at the specified position. This velocity is constant for an isotropic sublayer but is equal to $V_{P0}\sqrt{1+2\delta}$ for a VTI sublayer, where $V_{P0}$ is the vertical P-wave velocity and $\delta$ is the Thomsen's delta.
2. We substitute the results from step 1 into the recursion 12 starting from $k=1$ with $r_0 = 0$ and end up with $r_n = dx_n/dh$.
3. We can evaluate the final second-order traveltime derivative at the surface from equation 11 with the layer derivatives of the $n$-th layer and $r_n$ from step 2.
4. NMO or time-migration velocity can then be found according to equations 6 and 7 by multiplying the the result from step 3 with the total one-way traveltime in the reference 1-D medium $T = \sum^n_{k=0} T_k$.