Spherical Splines: beyond Spherical Harmonics for Non-uniform Geophysical Datasets on the Moon and Mercury

Introduction: Global-scale datasets, preferably with nearly uniform coverage are important elements of many geophysical studies of the planets and their moons. However, uniformly resolved data are not always available, for example, determination of the gravity field of the Earth’s moon suffers from this problem because the data depend upon line-of-sight tracking of a spacecraft’s radio signal. Due to the synchronous rotation of the Earth’s moon, direct determination of the far side gravity field is not possible with a single spacecraft resulting in a dataset with highly spatially variable quality [1]. An analogous situation exists where particular spacecraft orbits (i.e., highly elliptical ones) may also provide non-uniform data coverage or resolution, especially if the data depend upon spacecraft altitude (e.g., gravity, magnetic fields). The necessarily elliptical orbit with periapse limited to the northern hemisphere of Mercury in the future orbital phase of the MESSENGER mission is an example of the latter situation. Higher surface spatial resolution geophysical data will be limited to the northern hemisphere. Spherical harmonics are commonly used to represent planetary geophysical datasets [e.g. 2-7]. Among the primary advantages of spherical harmonics are their compact and convenient form as well as the availability of powerful spectral analysis techniques. In contrast to the ideal case, highly non-uniform data distributions reveal disadvantages associated with spherical harmonic representations. Indeed, they best represent data only for an expansion up to a degree and order appropriate for the data that are least resolved. This essentially truncated expansion (relative to the areas with high data resolution) introduces aliasing that results in an inaccurate representation of the data. Further, the relative truncation of the expansion amounts to discarding data from the more resolved regions. Unfortunately, spherical harmonics are relatively inflexible to local variations in data resolution. The global support basis for spherical harmonics is one source of their disadvantages with highly nonuniform data. However, non-uniformly resolved data may be more accurately represented by a model representation with local support. Local bases, especially those that rely on meshes produced by Delaunay triangulation, are adaptable to large variations in resolution [8]. Notably, interpolation support can be concentrated where resolving power is highest. Furthermore, locally supported basis functions can accommodate nonuniform, incomplete, and regional data distributions [8]. Spherical Basis Splines: Spherical basis splines (B-Splines) have been increasingly used to solve geophysical problems [see 9-12], particularly in seismology. A 2-D model on a sphere with a local cubic Bspline basis is parameterized in terms of geodesic distance on a triangular grid of knot points. The knot points are control points through which the piecewise polynomial (i.e. the spline) must pass. The normalized cubic B-spline functions are centered on N knots points i=1, 2,..., N on the surface of the planet. Each basis function fi( ,φ) depen s on the distance Δ from the ith knot point θ d with 3 4 ∆ ∆ ⁄ 3 2 ∆ ∆ ⁄ 1, ∆ ∆