First, the shot gathers are sorted into CMP
gathers. Therefore, shot interpolation becomes a trace interpolation problem
in the CMP domain. The traces to be interpolated are replaced with
zeroed traces. Let's call a given CMP gather to be
interpolated. The goal is to find a model space
that minimizes the
difference
between the known data and the modeled data with
an HRT operator
, i.e.,
![]() |
(65) |
![]() |
(66) |
![]() |
(67) |
In most cases, the data to be interpolated are aliased, creating
strong artifacts in . An efficient way to
mitigate these artifacts is by introducing a regularization operator
in equation (
) that will enforce sparseness in the
model space. This regularization isolates the strongest events
in the model space while ignoring the weakest.
To achieve this, a Cauchy regularization Sacchi and Ulrych (1995); Trad et al. (2003)
is introduced in equation (
) as follows:
![]() |
(68) |
By definition, introducing the Cauchy norm in equation
() makes the problem of finding
non-linear. To take the nonlinearity of the objective
function into account,
is minimized with the quasi-Newton method
introduced in Chapter
. In practice, this choice
leads to satisfying results after a few number of iterations (
).
Once a model is estimated, the interpolated CMP gathers
are obtained by forward modeling the data from the estimated
model space
and replacing the modeled traces
by the known traces as follows:
![]() |
(69) |
In the next section, the interpolation technique is applied to the data. This example illustrates that the shots can be effectively interpolated with the radon-based technique.