Collaborative Research: CMG: Multi-Resolution Inversion of Tectonically Driven Spatio-Temporal Gravity Signals Using Wavelets and Satellite Data

这项调查的主要目标包括对复杂地球物理过程产生的构造驱动的时空信号进行跨学科研究。这些过程包括收敛的板块边界、地震形变旋回、地幔对流、板内变形和冰川均衡调整(GIA)。目前，这些过程产生了表面变形形式的小但可测量的信号，目前只能通过使用全球定位系统的点测量或使用InSAR在小空间尺度(100公里)上探测到。这些“缓慢形变”信号的空间尺度从数百公里到大陆和行星尺度，时间尺度从一年到几十年，甚至几千年。地球重力场及其时空变化提供了对地球系统内综合质量再分布的洞察，代表了直接研究以多个自由度驱动这些复杂过程的机制的独特的基本可测量量。CHAMP、GRACE和GOCE等专门的卫星重力任务预计将首次以综合质量变化和垂直形变的形式测量这些小的、大范围的构造驱动信号。然而，当代表示地球位势的数学函数是传统的球谐函数，这不允许空间局部化。与球谐函数相比，三维小波具有显著的优势，例如在多分辨率表示和局部化方面，但它必须满足所谓的边值问题。总体科学目标是开发基于多分辨率的三维小波工具，以增强受构造驱动的时空重力信号，以改进分析，并在解决主要的公开科学问题方面取得进展，即了解陆地、海洋和冰盖表面上这些“缓慢变形”的驱动机制。研究人员建议开发基于两种小波方法的数学工具：(1)旋转不变球面小波函数；(2)不可分离的紧支撑张量积小波，用于表示时空重力场信号并执行地球物理“反转”以增强信号。这些重力信号的“反演”代表着严峻的数学和数值挑战，特别是考虑到需要多分辨率表示来增强局部化信号并考虑将小波扩展到包括时间维度。更广泛的影响和预期成果包括开发能够解决边值问题和利用卫星数据反演重力信号的三维小波工具，并在纳斯卡和南美板块地区演示和应用这项技术。所开发的数学工具旨在成为“普及”3-D小波用于教学和研究的第一步，并适用于许多跨学科的科学研究和工程问题。

The primary objectives of this investigation include interdisciplinary studies of tectonically driven spatio-temporal signals resulting from complex geophysical processes. These processes include convergent plate boundaries, earthquake deformation cycle, mantle convection, intra-plate deformations and Glacial Isostatic Adjustment (GIA). At present, these processes generate small but measurable signals in the form of surface deformations, which at present can only be detected over land by either point measurements using GPS, or on small spatial scales (100 km) using InSAR. These "slow deformation" signals have spatial scales longer than hundreds of km to continental and planetary scales, and temporal scales of a year to decades, and millennia. The Earth's gravity field and its spatio-temporal variations, providing insight on the integrated mass redistributions within the Earth's systems, represent a unique fundamental measurable quantity to directly study mechanisms which drive these complex processes with many degrees of freedom. For the first time ever, dedicated satellite gravity missions like CHAMP, GRACE and GOCE are anticipated to measure these small, broad-scale tectonically driven signals in the form of integrated mass change and vertical deformations. However, the contemporary mathematical functions to represent the geopotential are conventionally spherical harmonics which do not allow spatial localization. 3-D wavelets have notable advantages over spherical harmonics, e.g. for multi-resolution representation and localization, however, would have to satisfy the so-called "boundary-value problem". The overarching scientific goal is to develop multi-resolution based 3-D wavelet tools to enhance the tectonically driven spatio-temporal gravity signals for improved analyses and to make progress towards addressing the major open scientific questions of understanding the driving mechanisms of these "slow deformation" over land, ocean and ice-covered surfaces. The investigators propose to develop mathematical tools based on two wavelet approaches: (1) the rotational invariant spherical wavelet function, and (2) the non-separable compactly supported tensor-product wavelets to represent the spatio-temporal gravity field signals and perform geophysical "inversions" to enhance the signals. The "inversion" of these gravity signals represents stringent mathematical and numerical challenges, especially in light of the need for multi-resolution representation to enhance localized signals and to consider extending wavelets to include the time dimension. The broader impacts and anticipated results include the development of 3-D wavelet tools capable of solving the boundary value problem and inversion of gravity signals using satellite data and to demonstrate and apply the technique in the Nazca and South American plate region. The developed mathematical tools are intended to be among the first steps to "popularize" the use of 3-D wavelets for teaching and research, and are applicable to numerous interdisciplinary scientific studies and engineering problems.