Collaborative Research: CMG: Wavelet-Based Unified Approach for Physical Feature Extraction, Large-Scale Visualization, and Modeling of Multiscale Geological Processes

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This three-year proposal is a collaboration between three researchers whose talents span geophysics,applied mathematics,computational .uid dynamics,and computer science.The main objective of thisproposal is to develop a uni .ed framework for multiscale analysis,physical and statistical feature extraction,construction of multi-component phase diagram maps,large-scale visualization and modeling of geologicalprocesses using wavelets.We plan to apply this uni .ed approach to the analysis and simulation of thefollowing geological problems:1.High Rayleigh number thermal and thermo-chemical convection in Earth mantle,2.Strong .eld geodynamo (both numerical simulations and geodetical satellite data),3.Other kinds of simulations or geophysical observations.Second generation wavelets will be used to investigate new algorithms aimed at coherent-incoherent structureextraction (i.e.,plumes and background thermal .uctuations)from large-scale numerical simulations,todevelop new model equations that describe their space-time evolution,taking into account the e .ect of theincoherent .eld via statistical modeling,and to develop new wavelet-based visualization tools for both featureextraction and construction of multi-component phase diagram maps.In the area of modeling and simulation second generation wavelets will be used to separate the wide rangeof scales of high-Rayleigh number geophysical .elds into their incoherent and coherent components.Modernstatistical tools,specialized to non-homogeneous processes,will be applied to the stochastic modeling of thee .ect of incoherent .elds on the evolution of coherent .elds.The governing equations will be solved using adynamically adaptive wavelet collocation algorithm.An adaptive grid will enable the coherent structures tobe resolved and their evolution followed e .ciently.Wavelet-based tools will be developed to rapidly extract and visualize the relevant features in geophysicaldata sets.Coherent structures will be extracted at multiple scales.These wavelets will be visualized usingpoint rendering techniques for fast visual displays without the need for data decompression.These tools willbe applied to the analysis of the aforementioned geophysical problems.Finally,a general purpose tool will be developed to e .ciently capture,represent and visualize the phaseboundaries of n -dimensional phase diagram maps.These boundaries represent ow-dimensional features,andpose a formidable challenge to the geoscientist.Intellectual Merit.The algorithms developed herein speci .cally address problems of deterministicand/or stochastic modeling of dynamically active scales that are not explicitly resolvable,when classicalmethods have failed to yield progress in predictive modeling.These techniques have application to transportequations in other .elds of science and engineering confronted with a wide range of spatial scales.Novemethods proposed in this research will provide new insights on the internal dynamics of geosystems,willenable researchers to analyze large data sets and extract lower-dimensional features.Nove fully automatedrobust strategy of compact delineation and visualization of complicated realistic phase diagrams and relatedin situ physical properties of multicomponent systems are envisioned to be used in a variety of geophysical.elds such as geochemistry and petrology.Broader Impact.This project will drive and promote progress in computation,geophysical .uid dynam-ics,nonlinear physics and in the general application of wavelets to geoscientists.This opportunity will allowstudents from statistics,applied mathematics,computer science and geophysics to work collaboratively onthese technologically and mathematically cutting-edge areas with applied emphasis in the geosciences.