Smoothing Three-Dimensional Manifold Data, with Application to Tectonic Fault Detection
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Smoothing Three-Dimensional Manifold Data, with Application to Tectonic Fault Detection Carlo Grillenzoni1
Received: 28 April 2015 / Accepted: 27 November 2015 / Published online: 7 April 2016 © International Association for Mathematical Geosciences 2016
Abstract Three-dimensional manifold data arise in many contexts of geoscience, such as laser scanning, drilling surveys and seismic catalogs. They provide point measurements of complex surfaces, which cannot be fitted by common statistical techniques, like kriging interpolation and principal curves. This paper focuses on iterative methods of manifold smoothing based on local averaging and principal components; it shows their relationships and provides some methodological developments. In particular, it develops a kernel spline estimator and a data-driven method for selecting its smoothing coefficients. It also shows the ability of this approach to select the number of nearest neighbors and the optimal number of iterations in blurring-type smoothers. Extensive numerical applications to simulated and seismic data compare the performance of the discussed methods and check the efficacy of the proposed solutions. Keywords Bandwidth selection · Kernel splines · Local averaging · Principal surfaces · New Madrid earthquakes · Sliced scanning · Stopping criteria
1 Introduction Three-dimensional point data arise in applied geosciences as measurements of static features (e.g., stratifications) or dynamic events (e.g., landslides) of the earth. These data consist of space-time coordinates and marks associated with the points; they are used for building topographic surfaces and depth maps, or detecting hidden structures. Typical data generators are: (i) laser scanning, to obtain digital elevation models; (ii) surveys drilling, to measure mineral concentration and dispersion of pollutants; and
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Carlo Grillenzoni [email protected] Institute of Architecture, University of Venice, S. Croce 1957, 30135 Venezia, Italy
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Math Geosci (2016) 48:487–510
(iii) earthquake catalogs, to build maps of seismic risk and detecting tectonic faults. Smoothing of point measurements is necessary both to reduce the noise and to extract significant features of space. In the aforementioned context, estimating curves or surfaces with interpolation techniques of geostatistics is not always possible. This is due to the manifold structure of latent surfaces, where to a value of the planar coordinates there correspond multiple values of the depth coordinate. This problem has been recently stressed by Caumon and Collon-Drouaillet (2014) as concerned tectonic faults detection and recumbent folds representation. Fitting seismic data to learn about the local structure of the earth’s faults is a common practice of applied geologists (Jordan 2014), but in the multidimensional space, this issue is mainly handled with interactive computer graphics on filtered data (Jordan 2014). Hence, in the presence of large amounts of observations, there is a need for automatic numerical methods, at least in the ea
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