Velocity model evaluation through Born modeling and migration: a feasibility study

Born modeling and migration

We now use the modeled areal source to generate a new data set via Born modeling. To do this, we define the simulated dataset $d'$ recorded at arbitrary receiver locations $\mathbf{x}_r'$ :

$$d'(\mathbf{x}_r',\omega) = \sum_{\mathbf{x}'}\sum_{\mathbf{h}} \G... ...mega) G(\mathbf{x}'+\mathbf{h},\mathbf{x}_r',\omega) m(\mathbf{x'},\mathbf{h}).$$ (3)

Here, $m$ is the reflectivity model (in our case, the initial image), and the $\Gamma$ term is defined as

$$\Gamma(\mathbf{x}_s,\mathbf{h},\omega) = \sum_{\mathbf{x}_s} S(\mathbf{x}_s) G(\mathbf{x}_s,\mathbf{x}'-\mathbf{h}, \omega),$$ (4)

where $S$ is as defined in equation 1.

Because the placement of the receiver locations in equation 3 can be arbitrarily determined, they do not necessarily need to be on the surface, like the original recorded data. Placing the receivers at depth can improve the efficiency of this method by providing the capability for target-oriented imaging; if a velocity model is well-determined down to a given depth, the synthesized data can be recorded below that depth, avoiding unnecessary computation. This has a similar effect to re-datuming the wavefields, an approach taken by Wang et al. (2008) in their fast image updating strategy.

Now that we have new source and receiver wavefields, we can produce an image using standard wave-equation migration techniques:

$$m'(\mathbf{x}',\mathbf{h}) = \sum_{\omega}G^*(\mathbf{x}'-\mathbf... ...}'_r}G^*(\mathbf{x}'+\mathbf{h},\mathbf{x}'_r,\omega) d'(\mathbf{x}_r',\omega).$$ (5)

It is important to note that the Green's functions in equation 5 can be computed using any velocity model, and not necessarily the same one used to generate the source and receiver wavefields in previous steps. This can allow for testing of multiple possible velocity models. Furthermore, since subsurface offset gathers are generated during the imaging, we now have a more quantitative means of judging the accuracy of these various models. This represents an advantage over methods such as beam migration that cannot provide this information. Unfortunately, the imaging procedure as written in equation 5 can also generate crosstalk artifacts, since areal source data is used. While various methods are available to help attenuate these artifacts (Tang, 2008; Romero et al., 2000), we restrict our examples in the next section to isolated points in the subsurface, spaced far enough apart to limit the effects of crosstalk.

flat-orig,dip-orig
Figure 2. Prestack depth migration images of (a) a flat reflector and (b) a reflector dipping at a $20^{\circ }$ angle. The images were migrated with a constant velocity 15% too slow compared to the true velocity, causing the noticeable artifacts and lack of focusing in the subsurface offset dimension.

Velocity model evaluation through Born modeling and migration: a feasibility study