DEFINITIONS OF DMO

Here I review the definitions of two DMO operators. The detailed derivations are given earlier by Zhang (1988).

Let P(t,y,h) denote a constant offset section, P_n(t_n,y,h) the section after normal moveout (NMO) corrections, and P₀(t₀,y,h) the section after the NMO and DMO corrections. Figure shows the coordinates of these sections.

 

dmogeo
Figure 1 The geometry of the experiment: common mid-point versus common depth-point.

NMO correction is a time-stretching operation:

 

P_n(t_n,y,h)=P(t,y,h), (1)

where t and t_n are related by the normal moveout equation. Hale relates the DMO-corrected section to the NMO-corrected section as follow:

 

P₀(t₀,y,h)=P_n(t_n,y,h). (2)

From Figure , it is clear that the events represented by the two sides of equation (2) are not reflected from the same depth point if the dip of the reflector is nonzero. Therefore, the operator obtained from this relation may not preserve the correct amplitudes of the reflected events. To solve this problem, I define the DMO operation by explicitly relating the events reflected from the common depth points, as follows:

 

P_d(t_d,y_d,h)=P_n(t_n,y,h). (3)

From this definition, one would expect that the DMO operator would have some desirable properties in its amplitude spectrum. And, indeed, as the next section shows, this DMO operator has a uniform amplitude spectrum with respect to dips and offsets.