Imaging in shot-geophone space

The DSR equation in midpoint-offset space

By converting the DSR equation to midpoint-offset space we will be able to identify the familiar zero-offset migration part along with corrections for offset. The transformation between $(g,s)$ recording parameters and $(y,h)$ interpretation parameters is

$$y = { g + s \over 2 }$$ (15)

$$h = { g - s \over 2 }$$ (16)

Travel time $t$ may be parameterized in $(g,s)$-space or $(y,h)$-space. Differential relations for this conversion are given by the chain rule for derivatives:

$${\partial t \over \partial g} = \ {\partial t \over \partial y} {\partial y \over \parti... ...( {\partial t \over \partial y} +\ {\partial t \over \partial h } \right)$$ (17)

$${\partial t \over \partial s} = \ {\partial t \over \partial y} {\partial y \over \parti... ...( {\partial t \over \partial y} -\ {\partial t \over \partial h } \right)$$ (18)

Having seen how stepouts transform from shot-geophone space to midpoint-offset space, let us next see that spatial frequencies transform in much the same way. Clearly, data could be transformed from $(s,g)$-space to $(y,h)$-space with (9.15) and (9.16) and then Fourier transformed to $( k_y , k_h )$-space. The question is then, what form would the double-square-root equation (9.13) take in terms of the spatial frequencies $( k_y , k_h )$? Define the seismic data field in either coordinate system as

$$U ( s, g ) \eq U' ( y , h )$$ (19)

This introduces a new mathematical function $U'$ with the same physical meaning as $U$ but, like a computer subroutine or function call, with a different subscript look-up procedure for $(y,h)$ than for $(s,g)$. Applying the chain rule for partial differentiation to (9.19) gives

$${ \partial U \over \partial s} = \ { \partial y \over \partial s} { \partial U' \over \pa... ...y } +\ { \partial h \over \partial s} { \partial U' \over \partial h }$$ (20)

$${ \partial U \over \partial g} = \ { \partial y \over \partial g} { \partial U' \over \pa... ...l y } +\ { \partial h \over \partial g} { \partial U' \over \partial h }$$ (21)

and utilizing (9.15) and (9.16) gives

$${ \partial U \over \partial s} = \ {1 \over 2 } \left( { \partial U' \over \partial y } -\ { \partial U' \over \partial h } \right)$$ (22)

$${ \partial U \over \partial g} = \ {1 \over 2 } \left( { \partial U' \over \partial y } +\ { \partial U' \over \partial h } \right)$$ (23)

In Fourier transform space where $\partial / \partial x$ transforms to $i k_x$, equations (9.22) and (9.23), when $i$ and $U = U'$ are cancelled, become

$$k_s = {1 \over 2 } ( k_y - k_h )$$ (24)

$$k_g = {1 \over 2 } ( k_y + k_h )$$ (25)

Equations (9.24) and (9.25) are Fourier representations of (9.22) and (9.23). Substituting (9.24) and (9.25) into (9.13) achieves the main purpose of this section, which is to get the double-square-root migration equation into midpoint-offset coordinates:

$${\partial \over \partial z} U = - i {\omega \o... ...y - v k_h \over 2 \omega } \right)^2 } \right] U$$ (26)

Equation (9.26) is the takeoff point for many kinds of common-midpoint seismogram analyses. Some convenient definitions that simplify its appearance are

$$G = { v k_g \over \omega }$$ (27)

$$S = { v k_s \over \omega }$$ (28)

$$Y = { v k_y \over 2 \omega }$$ (29)

$$H = { v k_h \over 2 \omega }$$ (30)

The new definitions $S$ and $G$ are the sines of the takeoff angle and of the arrival angle of a ray. When these sines are at their limits of $\pm 1$ they refer to the steepest possible slopes in $(s,t)$- or $(g,t)$-space. Likewise, $Y$ may be interpreted as the dip of the data as seen on a seismic section. The quantity $H$ refers to stepout observed on a common-midpoint gather. With these definitions (9.26) becomes slightly less cluttered:

$$\begin{tabular}{\vert c\vert} \hline $\displaystyle {\st... ... + \sqrt{1-(Y-H)^2} \right) U$ \\ \hline \end{tabular}$$ (31)

EXERCISES:

1. Adapt equation (9.26) to allow for a difference in velocity between the shot and the geophone.
2. Adapt equation (9.26) to allow for downgoing pressure waves and upcoming shear waves.

Imaging in shot-geophone space