Multiwindow and multiwavelet methods in 1D, 2D and on the sphere

Wednesday, April 14, 2004, 2-3:00pm, WWH 1314

In geophysics, potential field data (gravitational anomalies and magnetic field variations) and topography are the primary sources of information on the internal structure of the planets. As one example, the elastic properties of the lithosphere (the rigid outer layer) can be estimated by comparing a planet's gravity signature with its topography by analyzing the cross-spectral properties of these fields. On the Earth, such calculations are usually performed in the Fourier domain on locally flat surfaces. On the Moon, Mars and Mercury, however, the effects of curvature become too important to neglect. Moreover spherical harmonic representations of the gravity field from satellite orbital determinations usually provide the primary data.

Adequate methods of spatio-spectral localization are required in order to extract spectral information from spatially localized regions. The joint optimization between spectral and spatial resolution is well studied on flat surfaces and has led to a variety of methods employing multiple data tapers or wavelets. In one dimension, D. Slepian found that prolate spheroidal wave functions on finite intervals are ideally concentrated in frequency. In two dimensions, I. Daubechies showed how weighted Hermite polynomials achieve a joint optimization in the time-frequency phase plane. We solve the analogous problem of finding optimally space-concentrated band-limited data windows on the sphere, using spherical harmonics expansions.

The solution of the associated eigenvalue problem leads to a new class of data windows that are mutually orthogonal on the sphere, and which can be calculated semi-analytically. We discusss their characteristics and point to their usage. The most concentrated window of the orthogonal family is shown to satisfy quantum-mechanical uncertainty principles of location (via the center of mass) and total angular momentum (via the angular degree l), and for the case where the angular order m=0, its properties are nearly identical to those of squeezed coherent states on the circle.