Linking Snell waves to Fourier transforms

To link Snell waves to Fourier transforms we merge equations () and () with equations (10)

$$\begin{eqnarray} {k_x \over \omega} \quad =\quad {\partial t_0 \over \partial x}... ... \cos \, \theta \over v } \quad =\quad{ \sqrt{1-p^2 v^2} \over v}\end{eqnarray}$$ (12) (13)

The basic downward continuation equation for upcoming waves in Fourier space follows from equation (7) by eliminating p by using equation (12). For analysis of real seismic data we introduce a minus sign because equation (13) refers to downgoing waves and observed data is made from up-coming waves.  

$$U( \omega , k_x ,z+\Delta z) \quad =\quad U( \omega , k_x ,... ...\over v} \ \sqrt{ 1 -\ {v^2k_x^2 \over \omega^2 }\ } \ \right)$$ (14)

In Fourier space we delay signals by multiplying by $e^{i\omega \Delta t}$,analogously, equation (14) says we downward continue signals into the earth by multiplying by $e^{i k_z \Delta z}$.Multiplication in the Fourier domain means convolution in time which can be depicted by the engineering diagram in Figure 5.

 

inout
Figure 5 Downward continuation of a downgoing wavefield.

Downward continuation is a product relationship in both the $\omega$-domain and the k_x-domain. Thus it is a convolution in both time and x. What does the filter look like in the time and space domain? It turns out like a cone, that is, it is roughly an impulse function of x²+z² - v² t². More precisely, it is the Huygens secondary wave source that was exemplified by ocean waves entering a gap through a storm barrier. Adding up the response of multiple gaps in the barrier would be convolution over x.

A nuisance of using Fourier transforms in migration and modeling is that spaces become periodic. This is demonstrated in Figure 6. Anywhere an event exits the frame at a side, top, or bottom boundary, the event immediately emerges on the opposite side. In practice, the unwelcome effect of periodicity is generally ameliorated by padding zeros around the data and the model.

 

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Figure 6 A reflectivity model on the left and synthetic data using a Fourier method on the right.