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.,
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:
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:
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.