Introduction

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 plane. 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). Claerbout (1999) suggests a straightforward least-squares optimization approach employing an iterative conjugate-gradient algorithm.

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 GMT software package.

Fomel et al. (1997) have recently shown that an iterative interpolation algorithm can be greatly accelerated by preconditioning with recursive multidimensional filters defined on a helix Claerbout (1998a,b). To construct a minimum-phase filter suitable for recursive filtering, one can apply an efficient spectral factorization method Sava et al. (1998).

In this paper, I develop an application of helical preconditioning to gridding with splines in tension. 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 interpolation 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).