The formulation of linear shaping regularization is (Fomel, 2007)

(1)

where
is a vector of model parameters; is the shaping operator; is the data; and and are the forward and adjoint operators respectively. In interpolation problems, is forward interpolation (in the case of irregular sampling) or simple masking (in the case of missing-data interpolation on a regular grid). In 1-D, shaping in Z-transform notation can be triangle smoothing (Claerbout, 1992)

(2)

for a given smoothing radius . One can visualize this as a convolution of two box filters producing a weighting triangle for a triangle / neighborhood radius of . Increasing produces a smoother model. In 2-D the shift operator translates into shifts along local slope. corresponds to PWD - which can be thought of as a differentiation - while its inverse operator
corresponds to PWC - similar to integration.

Seismic data interpolation using plane-wave shaping regularization