Moveout/radial coordinates in geophone space

Our theoretical analysis will abandon the geophone axis g in favor of a radial-like axis characterized by a Snell parameter p. (This really says nothing about the implied data processing itself, since it would be simple enough to transform final equations back to offset). The coordinate system being defined will be called a retarded, moveout-corrected, Snell trace, coordinate system. Ideal data in this coordinate system in a zero-dip earth is unchanged as it is downward continued. Hence the amount of work the differential equations have to do is proportional to the departure of the data from the ideal. Likewise the necessity for spatial sampling of the data increases in proportion to the departure of data from the ideal. Define

$$\sin \theta)/v$$ p

t_p any one-way time from the surface along a ray with parameter p

g the surface separation of the shot from the geophone

t' one-way time, surface to reflector, along a ray

$$\tau$$ travel-time depth of buried geophones, one-way time along a ray

t travel time seen by buried geophones

v(p, t_p)   a stratified velocity function v'(z), in the new coordinates

The coordinate system is based on the following simple statements: (1) travel time from shot to geophone is twice the travel time from shot to reflector, less the time-depth of the geophone; and (2) the horizontal distance traveled by a ray is the time integral of $v\,\sin\,\theta$ = pv²; (3) the vertical distance traveled by a ray is computed the same way as the horizontal distance, but with a cosine instead of a sine.

$$\begin{eqnarray} t(t' ,p, \tau ) \ \ \ &=&\ \ \ 2\ t' \ -\ \tau \\ g(t' ,p, \tau... ...int_0^{\tau} \ \ {v(p,t_p )} \ \sqrt { 1 \ -\ p^2 \, v^2}\ {dt_p}\end{eqnarray}$$ (8) (9) (10)

Surfaces of constant t' are reflections. Surfaces of constant p are rays. Surfaces of constant $\tau$ are datum levels. Unfortunately, it is impossible to invert the above system explicitly to get $(t' , p, \tau )$ as a function of (t,g,z). It is possible, however, to proceed analytically with the differentials. Form the Jacobian matrix  

$$\left[ \matrix { \matrix { \partial_{{t}' } \cr \partial_p \... ...\matrix { \partial_t \cr \partial_g \cr \partial_z } } \right]$$ (11)

Performing differentiations only where they lead to obvious simplifications gives the transformation equation for Fourier variables:  

$$\left[ \matrix { \matrix { - \omega' \cr k_p \cr k_{\tau}} ... ...\left[ \matrix { \matrix { - \omega \cr k_g \cr k_z} } \right]$$ (12)

It should be noted that (12) is a linear relation involving the Fourier variables, but the coefficients involve the original time and space variables. So (12) is in both domains at once. This is useful and valid so long as it is assumed that second derivatives neglect the derivatives of the coordinate frame itself. This assumption is often benign, amounting to something like spherical divergence correction.

Here we could get bogged down in detail, were we to continue to attack the nonzero offset case. Specializing to zero offset, namely, $p\,=\,0$, we get  

$$\left[ \matrix { \matrix { - \omega' \cr k_p \cr k_{\tau}} ... ...\left[ \matrix { \matrix { - \omega \cr k_g \cr k_z} } \right]$$ (13)

Equation (13) may be substituted into the single-square-root equation for downward continuing geophones, thereby transforming it to a retarded equation in the new coordinate system.