Velocity Independence

Forel (1988) shows how the DMO process can be split in a velocity independent shrinking of the input trace and a standard NMO step.

First an array of zero offset locations has to be designed. Each nonzero offset trace contributes to all zero offset traces whose combined source-receiver location lies in between the nonzero offset source and receiver. b is the distance between zero offset location and the midpoint of the nonzero offset experiment.

In the first step the nonzero offset trace time t_n is transformed into t₁:  

$$t_{n} = {t_{1}{h\over{k}}},$$ (16)

where the artificial offset k is defined according to (1)

 

k² = h²-b²

The new trace is added into an offset panel (k,t₁) at the zero offset location. Substituting (A-7), (A-8) into (A-6) yields the NMO relationship (3) for (t₀,t₁):

$$t_{0}^{2} = t_{1}^{2} - {4 k^{2}\over{V^{2}}}$$ (18)

After processing all nonzero offset traces a standard normal move out of the (k,t₁) panels results into a zero offset data set. The final NMO step permits a traditional velocity analysis. Unlike conventional NMO processing, DMO does not assume any particular reflector dip.

The equation set (A-3), (A-4) and (A-5) can be solved for (x,z) instead of (tn,tz). The points P(x,z) which fulfill the described conditions for varying traveltimes are called the locus of constant replacement point (lcr). Each P is the actual tangential point of an ellipse t_n and a circle centered at a fixed B. The two-way traveltime t(PB) is t₀. The traveltime t(SPR) is t_n. The lcr is a circle:  

$${(x - x_{0})}^{2}+z^{2} = {r}^{2}\\ x_{0} = {h^{2}+b^{2}\over{b}}\\ r = {{h^{2}-b^{2}}\over{2 b}}$$ (19)

Center location and radius are time independent.