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

Generalized source function

In conventional modeling and migration, a simple wavelet or plane wave is often used as the source function. However, here we can take advantage of the fact that the procedure described above begins with a migrated image. This allows us to perform post-stack ``exploding reflector" (Claerbout, 2005) modeling of a reflector or point diffractor in the subsurface; the upward-continued wavefield can be recorded at any location, and then injected as an areal source function during Born modeling. Mathematically, this areal source is described as

$$S(\mathbf{x}_s) = \sum_{\mathbf{x'}} \sum_{\mathbf{h}} G^*(\mathbf{x'}-\mathbf{h},\mathbf{x}_s,\omega,\mathbf{\xi}(\mathbf{x'},\mathbf{h})),$$ (1)

where $\mathbf{x}_s=(x_s,y_s,z_s)$ are the arbitrarily defined locations where the wavefield will be recorded; $\mathbf{h}$ is the vector of subsurface half-offsets; $\omega$ is angular frequency; $\mathbf{\xi}$ is the location of the exploding image point in the subsurface; and $G$ is the Green's function connecting the source to the image point (here, $^*$ denotes the adjoint). The Green's function is computed using the same velocity model that was used to image the originally-recorded data, meaning that the recorded wavefield should be independent of the original velocity model choice. However, since this velocity model is unlikely to be correct, the initial image should contain valuable information about the accuracy of this model in the form of subsurface offset gathers. Thus, the inclusion of the subsurface offset term $\mathbf{h}$ in equation 1 is designed to incorporate this information into the modeling. Since post-stack modeling is used to upward continue the wavefields, the non-zero subsurface offset data are mapped to equivalent zero subsurface offset locations:

$$\alpha(\mathbf{x-h})=\alpha(\mathbf{x-h}) + \beta(\mathbf{x,h}),$$ (2)

where $\alpha$ is the zero-offset data that are upward continued, and $\beta$ is the original subsurface offset gather. To illustrate the advantage gained by incorporating this information, Figures 1(a) and 1(b) show two recorded source wavefields from an image point that is actually located at $z=1000$ in the subsurface, but was initially imaged with a velocity that was 15% too slow. Both recorded wavefields have been reverse-propagated back to zero time to facilitate comparison. The source function in panel (a) was modeled using only the zero subsurface offset $h=0$ data from the initial image, while the result in panel (b) uses the non-zero offset information as written in equation 1. When only zero subsurface offset data are used, the source appears to focus at the incorrect depth; when the nonzero offset data are used, the effects of using the wrong velocity are apparent. Using the source function in Figure 1(b) should therefore prove superior for use with the Born modeling and migration scheme described in the next section.

pt-0,pt-n0
Figure 1. Recorded source wavefields that have been reverse-propagated to zero-time; the result in (a) does not include information from the nonzero subsurface offsets of the initial image, while (b) does include this information. Both the initial migration and the modeling used a velocity model that was 15% too slow.

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