Collaborative Research: A Field Expansion Method for Acoustic Scattering from Topography: Extensions to Elasticity and the Inverse Problem

这一提议启动了一项新的合作，旨在通过将地形纳入地震波建模并减少这些方法对极低频率数据的依赖来改进波形反演方法。首席研究人员建议调整和改进为模拟衍射光栅的散射而开发的方法。他们的方法扩展到线弹性，并提供了求解反问题的一般改进。为这一应用开发健壮而高效的技术需要有重大的数学进步。边界摄动法将场展开法扩展到任意数目的具有独立地形的层，并允许快速而准确地模拟声波在二维中的传播。需要扩展到三维和一般的弹性方程。此外，将基于几何声学的频率无关离散推广到多层弹性模型是一项重大而必要的数学进步。此外，为了使该技术真正适用于地震成像问题，反问题方面的进展也是必要的。“全波反演法”的标准方法要求数据的频率太低，无法在实际中记录。PI的方法将正演问题转化为一系列与地形有关的运算符的应用，其中界面形状看起来相当明确。对于这些形状的恢复，提出了一些迭代方案，使用结合病态问题理论中的标准正则化技术的组合的重新排列。地震波在沉积层中的传播特性在许多技术中都是至关重要的，包括确定地球内部的性质和结构、地震探测和预测以及油气勘探。鉴于它的许多重要应用，大量的数值和实验技术被用来解决这个问题也就不足为奇了。然而，在理解和能力方面仍然存在一些差距。PI通过复杂的数值模拟解决了其中的一些问题，这些数值模拟将通过实验室实验和青藏高原的现场测量进行验证。

This proposal initiates a new collaboration aimed at improving methods of waveform inversion by including topography in seismic wave modeling and reducing the reliance of these methods on data at extremely low-frequency. The Principal Investigators propose to adapt and improve methods developed for simulating scattering from a diffraction grating. Their approach has extensions to linear elasticity and provides general improvements for solving the inverse problem. Significant mathematical advances are required to develop robust and efficient techniques for this application. The Boundary Perturbation Method extends the Field Expansion approach to an arbitrary number of layers with independent topographies and allows for rapid and accurate simulation of acoustic wave propagation in two dimensions. Extensions to three dimensions and to the general equations of elasticity are required. Moreover, the extension of frequency-independent discretizations based on Geometric Acoustics to multilayered elastic models is a significant and necessary mathematical advance. Additionally, advances in the inverse problem are also necessary to make the technique truly applicable to the seismic imaging problem. The standard method of "full-waveform inversion" requires data at frequencies which are too low to record in practice. The PIs' approach casts the forward problem as the application of a sequence of topography-dependent operators where the interface shapes appear rather explicitly. A number of iteration schemes are proposed for the recovery of these shapes, using re-arrangements of the compositions coupled to standard regularizing techniques from the theory of ill-posed problems. The propagation properties of seismic waves in layers of sediment are crucial in many technologies including the determination of inner earth properties and structure, earthquake detection and prediction, and hydrocarbon (oil and gas) exploration. In light of its many important applications, it is not surprising that a vast array of numerical and experimental techniques have been brought to bear upon this problem. However, several gaps in understanding and capability still exist. The PIs' address some of these questions through sophisticated numerical simulations which will be validated against both laboratory experiments and field measurements from the Tibetan plateau.