Measuring flexural rigidity over individual geological provinces: On the design of optimal windows for spectral data analysis on flat two-dimensional domains

The effective elastic thickness of the lithosphere, its spatial variations, and its possible directional dependence, are geophysical quantities that provide much important information about the structure and evolution of the continents -- especially when combined with other observables such as heat flow, Moho depth, seismic anisotropy, and crustal age. The decades-old picture from the study by Bechtel et al. (Nature, 1990), that elastic thickness in North America is controlled largely by the thermal state of the lithosphere, is due for a revision, and for two reasons. The first is the new wealth of geophysical data being collected from the EarthScope suite of experiments; the second is substantial improvements in the measurement and modeling of gravity and topography and the inversion of their cross-power-spectral properties for flexural rigidity, including its anisotropy. In this presentation we focus on the latter, namely on the development of a new dedicated method to map elastic thickness variations in complex geologic terrains. To reach our goal of matching the tremendous data quality available for North America with measurement and modeling sophistication as far as elastic thickness is concerned, we initially focus on the somewhat more abstract problem of measuring wavelength-dependent properties (such as power spectra, admittance or coherence functions) of data collected over two-dimensional (2-D) geologic provinces of arbitrary description. Thomson's multi-taper method, which uses spatio-spectrally concentrated "Slepian" data windows has been widely used for estimating the spectral characteristics of geological data (gravity and magnetic anomalies, topography, etc). However, these "traditional" 2-D tapers suffer from problems both in the space and spectral domain. In the space domain, they have been restricted to the estimation of power spectra for data defined on rectangular regions. It is clear, however, that typical geologic provinces have irregular boundaries. In the spectral domain, the power spectra of the tapers themselves are not isotropic; this introduces anisotropy as artifacts of the measurement. In recent work, Simons, Wieczorek, and Dahlen have solved the spatiospectral concentration problem on the surface of the sphere; the resulting "Slepian" tapers are ideal to measure power spectral properties on irregularly shaped spherical domains. Here, by analogy, we have solved the Slepian concentration problem in a 2-D flat Cartesian domain: the solutions of this problem yield the tapers suitable for a true 2-D multitaper method, and allow the analysis of 2-D regions of arbitrary geometry (in the space domain); with a spectral response that is nevertheless isotropic (in the spectral domain). We illustrate the theory and numerical implementation of this new methodological development in the analysis of geophysical data by applying it to the study of anisotropy in the elastic thickness of the continental lithosphere, with an initial focus on the North American lithosphere.