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

$$\begin{eqnarray} y\ \ &=&\ \ {g \ +\ s \over 2 } \\ h\ \ &=&\ \ {g \ -\ s \over 2 }\end{eqnarray}$$ (37) (38)

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:

$$\begin{eqnarray} {\partial t \over \partial g} \ \ \ \ &=&\ \ \ \ {\partial t \... ... \over \partial y} \ -\ {\partial t \over \partial h } \, \right)\end{eqnarray}$$ (39) (40)

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 (37) and (38) and then Fourier transformed to ( k_y , k_h )-space. The question is then, what form would the double-square-root equation (35) take in terms of the spatial frequencies ( k_y , k_h )? Define the seismic data field in either coordinate system as  

$$U ( s, g )\ \eq \ U' ( y , h )$$ (41)

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 (41) gives

$$\begin{eqnarray} { \partial U \over \partial s} \ \ \ \ &=&\ \ \ \ { \partial y... ...{ \partial h \over \partial g}\ { \partial U' \over \partial h \ }\end{eqnarray}$$ (42) (43)

and utilizing (37) and (38) gives

$$\begin{eqnarray} { \partial U \over \partial s }\ \ \ \ &=&\ \ \ \ {1 \over 2 }... ...partial y \, }\ +\ { \partial U' \over \partial h \, } \, \right)\end{eqnarray}$$ (44) (45)

In Fourier transform space where $\partial / \partial x$ transforms to i k_x, equations (44) and (45), when i and U = U' are cancelled, become

$$\begin{eqnarray} k_s\ \ \ \ &=&\ \ \ \ {1 \over 2 }\ ( k_y\ -\ k_h ) \\ k_g\ \ \ \ &=&\ \ \ \ {1 \over 2 }\ ( k_y\ +\ k_h )\end{eqnarray}$$ (46) (47)

Equations (46) and (47) are Fourier representations of (44) and (45). Substituting (46) and (47) into (35) achieves the main purpose of this section, which is to get the double-square-root migration equation into midpoint-offset coordinates:  

$${\partial\ \over \partial z} \ U\ \ =\ \ -\,i \, {\omega \o... ..._y \,-\, v k_h \over 2\,\omega } \, \right)^2 \ } \ \right] \ U$$ (48)

Equation (48) is the takeoff point for many kinds of common-midpoint seismogram analyses. Some convenient definitions that simplify its appearance are

$$\begin{eqnarray} G\ \ \ \ &=&\ \ \ \ { v\ k_g \over \omega } \\ S\ \ \ \ &=&\ \ ... ...over 2\ \omega } \\ H\ \ \ \ &=&\ \ \ \ { v\ k_h \over 2\ \omega }\end{eqnarray}$$ (49) (50) (51) (52)

Chapter showed that the quantity $v \, k_x / \omega$ can be interpreted as the angle of a wave. Thus 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 (48) becomes slightly less cluttered:  

$$\begin{tabular} {\vert c\vert} \hline \\ $\displaystyle {\... ...\sqrt{1-(Y-H)^2} \ \right) U$\space \\ \\ \hline\end{tabular}$$ (53)

Most present-day before-stack migration procedures can be interpreted through equation (53). Further analysis of it will explain the limitations of conventional processing procedures as well as suggest improvements in the procedures.

EXERCISES:

1. Adapt equation (48) to allow for a difference in velocity between the shot and the geophone.
2. Adapt equation (48) to allow for downgoing pressure waves and upcoming shear waves.