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Estimating time shifts is a two step procedure where local stepouts are first
estimated and then integrated. The goal of dip estimation is to find
a local stepout, *p*_{h}, that destroys the
local plane wave such that,
| |
(1) |

where *u* is the wavefield at time , midpoint *x* and offset *h*.
For all gathers, we evaluate the slope *p*_{h} with
a method based on high-order
plane-wave destructor filters Fomel (2002). This technique
has the advantage of being accurate for steep dips. The estimation
of *p*_{h} is based on a non-linear algorithm (Gauss-Newton method) that
includes a regularization term. This regularization smooths dips
across offset and CMP location.
One problem with the current technique is that the dip
estimation algorithm cannot properly handle conflicting dips.
This can be troublesome when, for instance, multiples are present in the data.
To solve this problem, Fomel (2002) and
Brown (2002) show how two dips can be
estimated. Then if multiple reflections are present, the stepouts
corresponding to the primaries are kept and those of the multiples are
rejected.
Dips estimates lead to a vector of local stepouts that are integrated one CMP at a time to obtain time shifts.
Dips are smoothed along both
spatial directions but not in time .
To enforce smoothness in the direction
Lomask and Guitton (2004) introduce a time
component to .
The relationship between the local time shift vector, and the local dip vector ((.)^{T} being the transpose) at a constant *x* is:

| |
(2) |

where and are the partial derivative in
, *h* respectively.
In practice, we choose and control the amount of
smoothness by introducing a trade-off parameter as follows Lomask and Guitton (2004):
| |
(3) |

Using equation (3) we wish to minimize
the length of a vector that measures the difference
between and as follows:
| |
(4) |

where ,
and .We then minimize the following objective function:
| |
(5) |

where is the *L*_{2} norm.
By increasing , the estimated time shifts become smoother in time.
We solve equation (5) analytically in the Fourier domain
Lomask (2003), which speeds up the estimation of :

| |
(6) |

where
and.The dip integration yields the desired time shifts
plus a constant, i.e., a DC frequency component.
The zero frequency component is removed by subtracting the near offset
panel from the other offsets. Therefore, the time shifts are a
measure of the moveout errors relative to the near-offset panel.
At the end of the dip integration process, we end up with a map of
time shifts, . These time shifts can be used to
flatten the CMP gathers without any velocity analysis.
Our goal, however, is to find an interval velocity
function consistent with the estimated time shifts using
a tomographic inversion procedure.

** Next:** Tomography
** Up:** Guitton et al.: Velocity
** Previous:** Introduction
Stanford Exploration Project

5/23/2004