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Given the geometry in Figure we have proved in Part 1 that
| |
(10) |

where *t*_{0} is the traveltime from CMP to the reflector and back,
*t*_{h} is the source-receiver traveltime, 2*h* 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 *t*_{n} in equation (11) we have
| |
(13) |

Let us consider a pressure field *p*(*t*_{h},*y*,*h*) recorded as a
function of nonzero-offset time *t*_{h}, 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*(*t*_{h},*y*;*h*=*h*_{0}).
For all the values of the offset *h* we have a 3-D field *p*(*t*_{h},*y*;*h*).
We define a new field *p*_{n} (*t*_{n},*y*,*h*) as

| |
(14) |

obtained by replacing the value of the constant-offset traveltime *t*_{h}
in *p*(*t*_{h},*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 *t*_{h} to *t*_{n}.
Next we define another field *p*_{0}(*t*_{0},*y*,*h*) as

| |
(15) |

obtained by replacing the value of the NMO corrected traveltime *t*_{n}
in *p*_{n}(*t*_{n},*y*,*h*) by its expression in equation (13).
Equation (15) is dip dependent as it contains the
variable .The new field *p*_{0}(*t*_{0},*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
*p*_{0}(*t*_{0},*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 *p*_{0}(*t*_{0},*y*,*h*) by *p*_{n}(*t*_{n},*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