Differential equations and Fourier transforms

The chain rule for partial differentiation gives  

$$\left[ \ \matrix { \partial_t \cr \partial_g \cr \partial_s... ... \partial_y \cr \partial_{{h}' } \cr \partial_{\tau} } \right]$$ (66)

In our usual notation the Fourier representation of the time derivative $\partial_t$ is $- i \omega$.Likewise, $\partial_{ t' }$ and the spatial derivatives $( \partial_y , \partial_{{h}' } , \partial_{\tau} ,$$\partial_g , \partial_s , \partial_z )$ are associated with $i( k_y , k_{{h}' } , k_{\tau} ,$k_g , k_s , k_z ). Using these Fourier variables in the vectors of (66) and differentiating (58), (59), (60), and (61) to find the indicated elements in the matrix of (66), we get  

$$\left[ \matrix { \matrix { 1 \cr { - p } \cr p \cr { {2 \, ... ... { { - \omega' } \cr k_y \cr k_{{h}' } \cr k_{\tau} } } \right]$$ (67)

Let S be the sine of the takeoff angle at the source and let G be the sine of the emergent angle at the geophone. If the velocity v is known, then these angles will be directly measurable as stepouts on common-geophone gathers and common-shot gathers. Likewise, on a constant-offset section or a slant stack, observed stepouts relate to an apparent dip Y, and on a linearly moved-out common-midpoint gather, stepouts measure the apparent stepout H'. The precise definitions are

$$\begin{eqnarray} S \ \ \ &=&\ \ \ { v \, k_s \over \omega }\ \ \ \ \ \ \ \ \ \ ... ...\ \ \ \ \ \ \ \ \ \ \ H' \eq { v \, k_{{h}' } \over 2 \, \omega }\end{eqnarray}$$ (68) (69)

With these definitions the second and third rows of (67) become

$$\begin{eqnarray} G \ \ \ &=&\ \ \ \ p \, v \ + \ Y \ + \ H' \eq Y \ + \ ( H' \ +... ... \ - p \, v \ + \ Y \ - \ H' \ \ =\ \ Y \ - \ ( H' \ + \ p \, v )\end{eqnarray}$$ (70) (71)

The familiar offset stepout angle H is related to the LMO residual stepout angle H' by H' = H -pv. Setting H' equal to zero means setting k_h' equal to zero, thereby indicating integration over h' , which in turn indicates slant stacking data with slant angle p. Small values of H' /v or $k_{{h}' } / \omega$ refer to stepouts near to p.