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 | Inverse B-spline interpolation |  |
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In the notation of Claerbout (1999), inverse interpolation amounts to a
least-squares solution of the system
where
is a vector of known data
at irregular
locations
,
is a vector of unknown function values
at a regular grid
,
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
We can rewrite (17-18) in the form that
involves only spline coefficients:
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.
Subsections
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 | Inverse B-spline interpolation |  |
![[pdf]](icons/pdf.png) |
Next: Spline regularization
Up: Fomel: Inverse interpolation
Previous: Seismic applications of forward
2014-02-15