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## ABSTRACT
As a condition for further generalization of the migration
to zero-offset in variable velocity media,
I develop the theory for 2-D migration to zero offset (MZO)
in constant velocity media,
starting from prestack migration in midpoint-offset coordinates.
At the end of this chapter I arrive at an integral formulation
for the MZO operator, analytically derived from the double square
root (DSR) prestack migration equation. The integral formulation for
the MZO is very similar in form to the DSR equation, suggesting
a generalization to variable velocity media via a phase-shift
algorithm. Further chapters deal with offset separation and
the depth variable |

Introducing the Double Square Root Equation The theory for the double square root (DSR) equation is discussed in detail in the first chapter of Yilmaz's (1979) thesis. Without going into mathematical detail I will sketch the path of the basic theory for obtaining the DSR migration equation in offset and midpoint coordinates starting from the wave equation. Readers familiar with the DSR equation can skip directly to the next section.

The scalar wave equation in a 2-D medium of constant density can be written as

(1) |

(2) |

(3) |

(4) |

For a constant velocity we write

(5) |

(6) |

(7) |

To find the solution to equation (7) we would need to
have two initial or boundary conditions. We only
have the pressure field at *z*=0 as a boundary condition,
but we can still solve the problem if we decide to resolve
only the upgoing waves, in other words to use the exploding reflectors
principle.

If we know the
pressure field (or wavefield) at a certain depth we can
propagate it forward in time or backward in time. We
can also propagate it up in depth (along the *z*-axis) or down.
To understand how we determine the propagation direction
we have to analyze the values and
sign of *k*_{z}.
The function

(8) |

(9) |

(10) |

Knowing the wavefield at any depth *z _{0}* we can
find the wavefield at any other depth

However for depth varying velocity *v*(*z*) we have *k*_{z}
approximately constant only
for small depth intervals ()
where we can consider the velocity constant.
Therefore equation (10) becomes

(11) |

There are several restrictions on the values of *k*_{z}. Equation
(6) has the solution (7) only for real
values of *k*_{z} which imposes the condition

(12) |

In the case of a seismic experiment with many shots and receivers
we can downward continue separately the shots and the receivers
to any depth level.
The total phase shift to the same depth
level *z* becomes the phase shift of the
shots plus the phase shift of the receivers.

(13) |

(14) |

Isolating the Zero-offset migration The basic concept for analytically deriving the MZO from prestack migration is to separate the latter into two processes:

- Migration to zero offset.
- Zero-offset migration.

We start with the constant velocity **prestack migration**
in offset-midpoint coordinates (Yilmaz, 1979) formulated as:

(15) |

(16) |

The constant velocity **zero-offset migration**
(Gazdag, 1978) can be formulated as:

(17) |

(18) |

In order to convert
equation (15) into a form similar to equation (17),
I use a change of variables from to such that after integrating over the
variable *k*_{h}, equation (15) will be transformed
into the form:

Using Hale's (1983) derivation, a new variable is introduced in order to isolate the zero-offset migration operator. The expression for the new variable is found by equating the dispersion relation for prestack migration to the dispersion relation of zero offset migration,

and squaring the two equations twice. This algebra is demonstrated in detail in Hale's thesis (1983), Appendix 3.A, and therefore I did not repeat it here. The final expression for the variable is found to be(19) |

(20) |

In order to isolate the zero-offset
migration operator, after substituting the expression of
given by equation (19) in the
prestack migration equation (15,
the order of integration is changed between and *k*_{h}.
The integration
boundaries have to be observed carefully
as they are modified after each change of variables
and integration order. However, for the sake of simplicity, I will
ignore in the following demonstration the integration
limits, which are discussed in Appendix D.

By substituting the variable
in equation (15)
with its new expression (19) as a function of ,and changing the integration order between and *k*_{h}
the prestack migration equation becomes

(21) |

(22) |

(23) |

The last equation in (21) is of course the zero-offset migration equation (17), the classic zero-offset downward continuation and imaging described by Gazdag (1978) or Stolt (1978). Equation (22) represents a way of obtaining the zero-offset section from prestack data in midpoint-offset coordinates.

So far the operations needed to obtain the zero-offset stacked section from the prestack field are:

- 1.
- Fourier transform the prestack field .
- 2.
- Remap (interpolate) the data field from into .
- 3.
- Multiply by the Jacobian.
- 4.
- Integrate over
*k*_{h}. - 5.
- Inverse Fourier transform .

MZO as phase shift
The interpolated field
in equation (22)
represents the values of the
field after remapping from to
. It is obtained by first Fourier transforming
the initial prestack field along all three
(time, midpoint and offset) axes:
,and second interpolating from to .As in Stolt migration (Popovici *et al.* 1993),
we can replace the two steps of

- 1.
- Fourier transform with even sampling in ,
- 2.
- interpolation from to ,

Formally we inverse Fourier transform in time equation (22) to have

(24) |

(25) |

(26) |

The only drawback so far to equation (26) is that it performs a Fourier transform and later a summation over the offset variable. I will show in the next chapter how the offset variable can be separated and as a result MZO can be applied to distinct common-offset sections. Once MZO is applied to separate common-offset sections I isolate the conventional NMO and DMO processes. I will further show how equation (26) can be applied to variable velocity media, via a phase shift algorithm similar to Gazdag migration, and PSPI or split-step.

REFERENCES

- Claerbout, J. C., 1985, Imaging the Earth's Interior: Blackwell Scientific Publications.

- Gazdag, J., 1978,
Wave equation migration with the phase shift method:
Geophysics,
**43**, 1342-1351.

- Gazdag, J., and Sguazzero, P., 1984,
Migration of seismic data by phase shift plus interpolation:
Geophysics,
**49**, 124-131.

- Hale, I. D., 1983, Dip-moveout by Fourier transform: Ph.D. Thesis, Stanford University.

- Popovici, A. M., Blondel, P., & Muir, F., 1993,
Interpolation in Stolt migration:
SEP-
**79**, 261-264.

- Stoffa, P. L., Fokkema, J. T., de Luna Freire, R. M.,
and Kessinger, W. P., 1990,
Split-step Fourier Migration:
Geophysics,
**55**, 410-421.

- Stolt, R.H., 1978,
Migration by Fourier transform:
Geophysics,
**43**, 23-48.

- Yilmaz, O.,1979, Prestack partial migration: Ph.D. thesis, Stanford University.

- Zauderer, E., 1989, Partial Differential Equations of Applied Mathematics: Wiley-Interscience.

A In this appendix I show that by writing the variable function of , the double square-root (DSR) phase used in prestack migration is transformed to a new form corresponding to the phase used for zero-offset migration. The transformation from to as defined in equation (19) is:

whereThe DSR phase is transformed from

to Hale (1983) in the Appendix A of his thesis proves an equivalent assertion, with a different logic. Comparing the DSR phase with the phase of the zero-offset migration (defined as a single square root), Hale finds the expression of which transforms the former into the latter. I was tempted to refer the reader to his appendix as an indirect proof, but decided to include a thorough derivation, for completeness.Using the identity

I rewrite the DSR phase as(27) |

Examine the expression under the second square root (SSR) in equation (27)

and substitute for the expression in .The expression under the second square root becomes:(28) |

(29) |

B The purpose of this appendix is to evaluate the Jacobian of the transformation from to :

Starting with the transformation of variable and differentiating we have and therefore the Jacobian is:C The purpose of this appendix is to find the inverse of the transformation , or to express function of .Starting with the original transformation of variable we have

and note that has always the same sign as .Square the equation to obtain and after isolating the terms in we have the equation(30) |

(31) |

(32) |

(33) |

D
In this appendix I follow the integration boundaries for
all the integral transformations from equation (15)
to equation (26).
In equation (15) the values of the constant *k*_{z},
given by equation (16),
have to be real. This requires the conditions

(34) |

In Figure the shaded area represents the
region of integration established by equation ((34)
for a constant *k*_{y}.
khkyomegaheight=2.in,width=6.in.
Regions of integration.
The existence condition for *v*_{h} in equation (34)
requires the integration boundaries in equation (15) to
be as follows:

Finally, the change of integration variable from back to from equation (24)
to equation (26) will restore the initial condition
for *k*_{h}:

5/15/2001