** Next:** Finite differences and spectral
** Up:** Regularizing smooth data with
** Previous:** Regularizing smooth data with

The traditional minimum-curvature criterion implies seeking a
two-dimensional surface *f*(*x*,*y*) in region *D*, which corresponds to
the minimum of the Laplacian power:

| |
(91) |

where denotes the Laplacian operator: .
Alternatively, we can seek *f*(*x*,*y*) as the solution of the biharmonic
differential equation

| |
(92) |

Equation () corresponds to the normal system of equations
in the least-square optimization problem (), the
Laplacian operator being , and the surface *f*(*x*,*y*)
corresponding to the unknown model . Fung (1965) and
Briggs (1974) derive equation () directly
from () with the help of the variational calculus and
Gauss's theorem.
Formula () approximates the strain energy of a thin
elastic plate Timoshenko and Woinowsky-Krieger (1968). Taking tension into account modifies
both the energy formula () and the corresponding
equation (). Smith and Wessel (1990) suggest the
following form of the modified equation:

| |
(93) |

where the tension parameter ranges from 0 to 1. The
corresponding energy functional is
| |
(94) |

Zero tension leads to the biharmonic equation () and
corresponds to the minimum curvature construction. The case of
corresponds to infinite tension. Although infinite tension
is physically impossible, the resulting Laplace equation does have the
physical interpretation of a steady-state temperature distribution. An
important property of harmonic functions (solutions of the Laplace
equation) is that they cannot have local minima and maxima in the free
regions. With respect to interpolation, this means that, in the case
of , the interpolation surface will be constrained to have
its local extrema only at the input data locations.
Norman Sleep (2000, personal communication) points out that if the
tension term is written in the form , we can follow an analogy with heat flow and
electrostatics and generalize the tension parameter to a
local function depending on *x* and *y*. In a more general form,
could be a tensor allowing for an anisotropic smoothing in
some predefined directions similarly to Clapp's steering-filter method
Clapp et al. (1997).

To interpolate an irregular set of data values, *f*_{k} at points
(*x*_{k},*y*_{k}), we need to solve equation () under the
constraint

| |
(95) |

which translates to equation () in the linear operator
notation. Using the results of Chapter , we can
accelerate the solution by recursive filter preconditioning. If
is the discrete filter representation of the differential
operator in equation (), and we can find a minimum-phase
filter whose autocorrelation is equal to , then
an appropriate preconditioning operator is a recursive inverse
filtering with the filter . Formulating the problem in
helical coordinates Claerbout (1998a,b) enables both
the spectral factorization of and the inverse filtering
with .

** Next:** Finite differences and spectral
** Up:** Regularizing smooth data with
** Previous:** Regularizing smooth data with
Stanford Exploration Project

12/28/2000