The basic target of seismic interpolation is to solve the following equation:
is the observed data which is regularly or irregularly sampled,
is the unknown data we want to reconstruct and
is the sampling matrix. The sampling operator has a diagonal structure, which is composed by zero and identity matrix:
in equation 2 corresponds to sampling a trace, and each
corresponds to missing a trace.
As equation 1 is under-determined, additional constraint is required in order to solve the equation. By applying a regularization term, we get a least-squares minimization solution for solving equation 1:
is a regularization operator and
denotes the square of norm.