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** Up:** Fomel: Inverse interpolation
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In the notation of Claerbout (1999), inverse interpolation amounts to a
least-squares solution of the system
| |
(15) |

| (16) |

where is a vector of known data *f*(*x*_{i}) at irregular
locations *x*_{i}, is a vector of unknown function values
*f*(*n*) at a regular grid *n*, is a linear interpolation
operator of the general form (1), is an
appropriate regularization (model styling) operator, and is
a scaling parameter. In the case of B-spline interpolation, the
forward interpolation operator becomes a cascade of two
operators: recursive deconvolution , which converts the
model vector to the vector of spline coefficients
, and a spline basis construction operator .System (15-16) transforms to
| |
(17) |

| (18) |

We can rewrite (17-18) in the form that
involves only spline coefficients:
| |
(19) |

| (20) |

After we find a solution of system (19-20),
the model will be reconstructed by the simple convolution
| |
(21) |

This approach resembles a more general method of model preconditioning
Fomel (1997a).
The inconvenient part of system (19-20) is the
complex regularization operator . Is it possible to avoid
the cascade of and and to construct a
regularization operator directly applicable to the spline coefficients
? In the following subsection, I develop a method for
constructing spline regularization operators from differential
equations.

** Next:** Spline regularization
** Up:** Fomel: Inverse interpolation
** Previous:** Seismic applications of forward
Stanford Exploration Project

9/5/2000