The Velocity Independent DMO Equation

 In the early 1980's G. H. F. Gardner and colleagues at the University of Houston showed that it was possible to DMO-correct recorded data in a velocity-independent manner. The fundamental basis for Gardner's method for velocity-independent DMO is the recognition that the double-square-root equation can be converted into a single-square-root equation by a radial-plane-diffraction process. After application of DMO, a second diffraction process, which takes place on constant time-slices completes the prestack migration step.

Consider a constant velocity medium with velocity v. With reference to Figure , the double-square root (DSR) equation is:  

$$T = \sqrt{T_0^2 + \frac{(m - m_0 + h)^2}{v^2}} + \sqrt{T_0^2 + \frac{(m - m_0 - h)^2}{v^2}}.$$ (1)

Here m is the midpoint between source S and receiver R, m₀ is the arbitrary location of a fixed scatter point, h is the half-offset, T₀ is the one-way zero-offset traveltime from the normal to the constant-time ellipse at (x = (m - m₀),y) to b, and T is the time required to traverse the path SPR. Note that in this case, both m₀ and m are referenced to the same fixed origin while x = m - m₀ is relative to m.

 

Fig1
Figure 1 Depth section showing constant-time ellipse for the DSR equation.

The ellipse in Figure is the locus of all points (x,y) for which T is constant. It will be convenient to write (1) in the form:

 

$$T = \sqrt{T_0^2 + \frac{(x + h)^2}{v^2}} + \sqrt{T_0^2 + \frac{(x - h)^2}{v^2}}.$$ (2)

Following Forel and Gardner (1988), let  

$$A = \frac{vT}{2},$$ (3)

and  

$$B = \sqrt{(A^2 - h^2)}.$$ (4)

Then  

$$\frac{x^2}{A^2} + \frac{y^2}{B^2} = 1,$$ (5)

 

$$b = \frac{4xh^2}{v^2T^2},$$ (6)

and the distance from b to P satisfies  

$$d^2 = B^2(1 - \frac{b^2}{h^2}).$$ (7)

Since  

$$d^2 = \frac{v^2T_0^2}{4} = (\frac{v^2T^2}{4} - h^2)(1 - \frac{b^2}{h^2}),$$ (8)

one has  

$$T_0^2 = (1 - \frac{b^2}{h^2})T^2 - \frac{4h^2}{v^2}(1 - \frac{b^2}{h^2})$$ (9)

This is the basic result from which DMO will be derived.