Statistics – Computation
Scientific paper
Dec 2004
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2004agufm.g33a..06w&link_type=abstract
American Geophysical Union, Fall Meeting 2004, abstract #G33A-06
Statistics
Computation
3299 General Or Miscellaneous, 1219 Local Gravity Anomalies And Crustal Structure, 1227 Planetary Geodesy And Gravity (5420, 5714, 6019), 0903 Computational Methods, Potential Fields, 0920 Gravity Methods
Scientific paper
One method of deciphering the near surface interior structure of a planet is through the joint analysis of its gravity and topography fields. Countless such studies have been undertaken for the Earth, and for a large portion of these, the analysis was performed in the spectral domain. Given the high spatial resolution of the measurements, and the generally small scale of the regions of interest in these studies, it was often appropriate to neglect the Earth's curvature and to utilize the assumption of Cartesian geometry. In doing so, many powerful methods become available to the analyst, such as multitaper spectral estimation methods and wavelet analysis techniques. In contrast, when analyzing the gravity fields of the other terrestrial planets, the assumptions of Cartesian geometry are no longer tenable. The primary gravity models for these bodies are often expressed in terms of global spherical harmonic coefficients, and these all have a much lower spatial resolution than that of the Earth. While localized spectral analysis techniques in Cartesian geometry have reached a certain level of maturity, analogous techniques in the spherical domain are still in their infancy. The purpose of this talk is to address the fundamental question of how one can obtain spectral estimates of a function expressed on a sphere that are localized to a certain "geographic" region. Using these localized spectral estimates, one can then calculate localized admittance and coherence functions, and then compare these with the predictions from a similarly localized model. Examples of this methodology will be applied to the planet Mars, showing how localized estimates of elastic thickness and load density can be obtained. The method described here is the spherical analog to Slepian's Cartesian concentration problem and Thomson's multitaper spectral estimation method. The first aspect of this method is to find a suitable windowing function in order to "localize" a geographic province on the sphere. By solving an optimization problem, a family of orthogonal functions are obtained. Next, localized spectral estimates can be obtained by multiply the data by these functions, and then expanding the resulting field in spherical harmonics. For the simple case when the input spectra is "white" we show that the localized spectral estimates are nearly unbiased. However, when the input spectra is "red" (which is the case of gravity of topography fields), the spectral bias can be significant. We show that the spectral estimates associated with a single localizing window are in general a poor approximation of the true localized spectra. However, by employing multiple data tapers, the multitaper spectral estimate becomes increasing robust as the number of windows increases.
Simons Frederik J.
Wieczorek Mark A.
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