Common-midpoint slanted wave analysis is a more conservative approach to seismic data analysis than the Snell wave approach. The advantage of common-midpoint analysis is that the effects of earth dip tend to show up mainly on the midpoint axis, and the effect of seismic velocity shows up mainly on the offset axis. Our immediate goal is to define an invertible, wave-equation approach to determination of interval velocity.
The disadvantage of common-midpoint analysis is that it is nonphysical. A slant stack at common geophone simulates a downgoing Snell wave, and you expect to be able to write a differential equation to describe it, no matter what ensues, be it multiple reflection or lateral velocity variation. A common-midpoint slant stack does not model anything that is physically realizable. Nothing says that a partial differential equation exists to extrapolate such a stack. This doesn't mean that there is necessarily anything wrong with a common-midpoint coordinate system. But it does make us respect the Snell wave approach even though its use in the industrial world is not exactly growing by leaps and bounds.
(Someone implementing common-midpoint slant stack would immediately notice that it is easier than slant stack on common-geophone data. This is because at a common midpoint, the tops of hyperboloids must be at zero offset, the location of the Fresnel zone is more predictable, and interpolation and missing data problems are much alleviated).
Seismic data is collected in time, geophone, shot, and depth coordinates (t,g,s,z). A new four-component system will now be defined. Midpoint is defined in the usual way:
Define the LMO time as the travel time in the point-source experiment less the linear moveout. So, at any depth, the LMO time is .As h' was defined to be the surface half-offset, t' is defined to be the surface LMO time. From the LMO time of a buried experiment, the LMO time at the surface is defined by adding in the travel-time depth of the experiment:
From the geometry of Figure 17 it will be deduced that a measurement of a reflection at some particular value of (h' ,t' ) directly determines the velocity. Write an equation for the reflector depth:
Gathering the above definitions into a group, and allowing for depth-variable velocity by replacing z by the integral over z, we get
Before these equations are used, all the trigonometric functions must be eliminated by Snell's law for stratified media, .Snell's parameter p is a numerical constant throughout the analysis.
The equation for interval velocity determination (50) again arises when dt' /dz from (58) and d h' /dz from (60) are combined:
At the earth's surface z= 0, seismic survey data can be put into the coordinate frame (58), (59), (60), and (61) merely by making a numerical choice of p and doing the linear moveout. No knowledge of velocity v(z) is required so far. Then we look at the data for some tops of the skewed hyperbolas. Finding some, we use equation (50), (57) or (63) to get a velocity with which to begin downward continuation.
Waves can be described in either the (t,g,s,z) physical coordinates or the newly defined coordinates .In physical coordinates the image is found at