Regularizing smooth data with splines in tension

The method of minimum curvature is an old and ever-popular approach for constructing smooth surfaces from irregularly spaced data Briggs (1974). The surface of minimum curvature corresponds to the minimum of the Laplacian power or, in an alternative formulation, satisfies the biharmonic differential equation. Physically, it models the behavior of an elastic plate. In the one-dimensional case, the minimum curvature method leads to the natural cubic spline interpolation de Boor (1978). In the two-dimensional case, a surface can be interpolated with biharmonic splines Sandwell (1987) or gridded with an iterative finite-difference scheme Swain (1976). According to the general optimization method, outlined in Chapter , I approach the gridding (data regularization) problem with an iterative least-squares optimization scheme.

In most of the practical cases, the minimum-curvature method produces a visually pleasing smooth surface. However, in cases of large changes in the surface gradient, the method can create strong artificial oscillations in the unconstrained regions. Switching to lower-order methods, such as minimizing the power of the gradient, solves the problem of extraneous inflections, but also removes the smoothness constraint and leads to gradient discontinuities Fomel and Claerbout (1995). A remedy, suggested by Schweikert (1966), is known as splines in tension . Splines in tension are constructed by minimizing a modified quadratic form that includes a tension term. Physically, the additional term corresponds to tension in elastic plates Timoshenko and Woinowsky-Krieger (1968). Smith and Wessel (1990) developed a practical algorithm of 2-D gridding with splines in tension and implemented it in the popular GMT software package.

In this section, I develop an application of helical preconditioning to gridding with splines in tension. Following the results of Chapter , I accelerate an iterative data regularization algorithm by recursive preconditioning with multidimensional filters defined on a helix Claerbout (1998a). The efficient Wilson-Burg spectral factorization constructs a minimum-phase filter suitable for recursive filtering.

I introduce a family of 2-D minimum-phase filters for different degrees of tension. The filters are constructed by spectral factorization of the corresponding finite-difference forms. In the case of zero tension (the original minimum-curvature formulation), we obtain a minimum-phase version of the Laplacian filter. The case of infinite tension leads to spectral factorization of the Laplacian and produces the known helical derivative filter Claerbout (1999); Zhao (1999).

The tension filters can be applied not only for data regularization but also for preconditioning in any estimation problems with smooth models. Tomographic velocity estimation is an obvious example of such an application Woodward et al. (1998).