Fractional Viscoacoustic Wave Equations: Mathematical Analysis, Efficient Simulations, and Applications to Full-Waveform Inversion of Seismic Data

地震波形反演是石油天然气工业和地球物理研究中广泛使用的一种技术。这项技术是最强大的地下成像技术之一；它的成功既依赖于准确的地震波传播模型，也依赖于实际实施的有效数值计算。该项目解决了这两个方面的基本数学和计算问题，因此对各种应用至关重要，包括地球内部成像、石油和天然气工业的地下勘探、碳封存和地热设施的监测以及地震工程。学生将通过参与跨学科的研究活动进行培训。本研究的主要目标是建立分数阶粘声波动方程的数学和计算处理，提供高效和准确的建模算法，并进一步构建地震数据的粘声全波形反演算法。分数阶粘声波动方程既考虑了地震波的衰减，又考虑了地震波传播过程中的频散，为研究衰减地震波开辟了一个新的方向。然而，它的非局部性带来了相当大的挑战。该项目将涉及数据反演、验证和应用的数学分析、数值模拟和地球物理应用方面的系统研究。在理论方面，我们将研究分数阶粘声波动方程的非局域性质。在数值方面，将为其有效的模拟开发数值分析和有效的算法。在这些结果的基础上，将解决与分数粘声波方程相关的粘声全波形反演的实现问题。全波形反演的验证将使用合成数据和德克萨斯州Frio地层二氧化碳封存现场的现场数据进行。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Seismic waveform inversion is a widely used technique in the oil and gas industry as well as in geophysical research. This technique is among the most powerful subsurface imaging techniques; its success relies on both an accurate model of seismic wave propagation and efficient numerical computation for the practical implementation. This project addresses fundamental mathematical and computational issues in these two aspects and thus is of central importance for various applications, including imaging of the Earth's interior, subsurface exploration in the oil and gas industry, monitoring in carbon sequestration and geothermal facilities, and earthquake engineering. Students will be trained through involvement in the interdisciplinary research activities.The main objectives of this research are to build mathematical and computational treatments for fractional viscoacoustic wave equations, to provide efficient and accurate modeling algorithms, and to further construct the viscoacoustic full-waveform inversion algorithm of seismic data. The fractional viscoacoustic wave equation, accounting for both seismic attenuation and dispersion in wave propagation, opens a promising direction to study seismic waves with attenuation. However, its nonlocality introduces considerable challenges. This project will involve systematic research in mathematical analysis, numerical simulations, and geophysical applications of data inversion, validation, and applications. On the theoretical side, the nonlocal properties of the fractional viscoacoustic wave equation will be investigated. On the numerical side, numerical analysis and efficient algorithms will be developed for its effective simulation. Building on these results, implementation issues for the viscoacoustic full-waveform inversion associated with fractional viscoacoustic wave equations will be addressed. The validation of the full-waveform inversion will be carried out with both synthetic data and field data from the Frio Formation carbon dioxide sequestration site in Texas.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.