Velocity-Independent DMO

 Now define a new offset h_g and time T₁, as follows:

 

h_g² = h² - b², (10)

and  

$$T_1^2 = T_0^2 + \frac{4h_g^2}{v^2}.$$ (11)

Then after rearranging (9), one has  

$$T_1 = \frac{h_g}{h}T,$$ (12)

Equations (10), (11), and (12) define velocity-independent DMO. They provide the framework for mapping traces with offset h and midpont m to zero-offset traces with offset h_g and midpoint m + b. Note that this process requires no velocity information and can be accomplished by a careful sampling or ``binning'' of b Gardner (1993). The trajectory of a single time sample T for a trace with half-offset h in the new offset space h_g is shown in Figure .

 

Fig2
Figure 2 Radial-plane DMO.

Each fixed sample in (m,h,t) space is mapped to a curve in (m+b,h_g,t) space. Here, the curve lies in a radial plane, within the data volume (m,h,t), defined by $h = \frac{h_g}{T_1}t$. It is possible to show Gardner (1993) that the mapping can be accomplished by (f,k) migration of the radial-plane data with the "velocity" $\frac{h_g}{T_1}$.Thus, the computational requirements for velocity-independent DMO are $O(n \log n)$. Although it is not within the scope of this article, Gardner 1993 also provides a log-stretch-(f,k) version of this DMO when h (common-offset section) is fixed.

After DMO, the new output volume consists of zero-offset but unmigrated data. Arrival times are specified by (11) and as a result are completely hyperbolic functions of h_g. The new data set can be NMO corrected, and stacked, to produce a zero-offset section. Any form of standard time migration may then be applied to produce a proper subsurface image.

Note also that if each new equivalent offset is migrated after NMO but prior to stack, then each point on the diffraction curve of Figure will result in one of the circles indicated in Figure . After stack, the envelope of the circles, as also indicated in Figure , will be the equal-traveltime ellipse of (2). Since the radii of each of these circles is given by the velocity-independent equation (7), one can postulate that constant-velocity-prestack imaging requires no initial velocity information.

 

Fig3
Figure 3 Ellipse as envelope of circles.