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

本次调查的主要目标包括跨学科的研究，构造驱动的时空信号产生的复杂的地球物理过程。 这些过程包括会聚板块边界、地震变形旋回、地幔对流、板内变形和冰川均衡调整。 目前，这些过程以表面变形的形式产生小但可测量的信号，目前只能通过使用全球定位系统的点测量或使用干涉合成孔径雷达的小空间尺度（100公里）在陆地上探测这种信号。 这些“缓慢变形”信号的空间尺度从几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。 地球的重力场及其时空变化，提供了对地球系统内的综合质量再分布的洞察力，代表了一个独特的基本可测量的量，可以直接研究驱动这些具有许多自由度的复杂过程的机制。 有史以来第一次，像CHAMP，GRACE和GOCE这样的专用卫星重力任务预计将以综合质量变化和垂直变形的形式测量这些小而大规模的构造驱动信号。然而，当代的数学函数来表示的地球势是传统的球谐函数，不允许空间定位。 3-D小波具有比球谐函数显著的优点，例如用于多分辨率表示和定位，然而，将必须满足所谓的“边界值问题”。首要的科学目标是开发基于多分辨率的三维小波工具，以增强构造驱动的时空重力信号，改进分析，并在解决主要的开放性科学问题方面取得进展，以了解这些“缓慢变形”在陆地、海洋和冰雪覆盖的表面上的驱动机制。 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，和（2）不可分离的压缩支持张量积小波来表示时空重力场信号并执行地球物理“反演”以增强信号。 这些重力信号的“反演”代表了严格的数学和数值挑战，特别是鉴于需要多分辨率表示以增强局部信号并考虑扩展小波以包括时间维度。 更广泛的影响和预期成果包括开发能够解决边界值问题的三维小波工具和利用卫星数据反演重力信号，并在纳斯卡和南美洲板块区域演示和应用这一技术。 开发的数学工具的目的是“普及”使用3-D小波的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.