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## Hale's DMO

Given the geometry in Figure we have proved in Part 1 that
 (10)
where t0 is the traveltime from CMP to the reflector and back, th is the source-receiver traveltime, 2h is the distance between source and receiver and v is the velocity of the medium. Note that in this situation the reflection point R in the nonzero-offset case differs from the actual reflection point S in the zero-offset case.

HaleDMO
Figure 12
Geometry for a dipping reflector in a constant velocity medium. Notice that the reflection point for the nonzero-offset ray R is different from the zero-offset reflection point S. The dipping angle is .

Hale (1984) uses equation (10) to write
 (11)
and we observe that the NMO corrected time is
 (12)
Substituting tn in equation (11) we have
 (13)

Let us consider a pressure field p(th,y,h) recorded as a function of nonzero-offset time th, midpoint y and offset h. In a constant-offset section we set the variable h to a constant value, so we have a 2-D field p(th,y;h=h0). For all the values of the offset h we have a 3-D field p(th,y;h). We define a new field pn (tn,y,h) as
 (14)
obtained by replacing the value of the constant-offset traveltime th in p(th,y,h) by its expression in equation (12)

Note that for a constant value of h this transformation amounts to shifting a value in a trace from th to tn.

Next we define another field p0(t0,y,h) as
 (15)
obtained by replacing the value of the NMO corrected traveltime tn in pn(tn,y,h) by its expression in equation (13). Equation (15) is dip dependent as it contains the variable .The new field p0(t0,y,h) so far is unknown and further computations are needed to determine it. However equation (15) formally represents a mapping from a NMO corrected field to a DMO corrected field. Remember again that in this formulation the nonzero-offset reflection point R does not correspond to the zero-offset reflection point S.

So far in equation (15) the only variable that we cannot easily determine is so we will try to find a transformation to express as a function of other variables. We have in a zero-offset section

as seen in Figure . In equation (9) we proved that for a dipping segment we have
 (16)

Now we need to Fourier transform the pressure field p0(t0,y,h) to take advantage of the new variables that we used in equation (16). We have
 (17)

We can use the mapping we defined in equation (15) and replace p0(t0,y,h) by pn(tn,y,h) in equation (17). By changing the variable of integration we need to calculate the Jacobian of the transformation (13). We have

and
 (18)
We can now rewrite equation (17) as
 (19)
which is Hale's DMO by Fourier transform.

Next: Zhang's improved DMO Up: DMO BY FOURIER TRANSFORM Previous: 2-D Fourier transforms of
Stanford Exploration Project
11/17/1997