基于高维最优传输距离目标函数的衰减介质全波形反演研究

Full waveform inversion (FWI) is convinced to be a more accurate velocity model building method, compared with the other methods. However, when the observed data is lack of effective low frequency components and the initial model is far from the true model, FWI suffers from a cycle-skipping problem. The newly developed optimal transport distance (OTD) objective function based FWI is able to suppress the cycle-skipping problem due to its good convexity, but still deserves a deep study because of some unresolved issues. Firstly, the greatest strength of the OTD objective function is to compare two high-dimensional seismic data as a whole. However, the current OTD based FWI compares seismic data trace by trace, or shot by shot, instead of comparing them as a whole, which loses the advantages of OTD. Secondly, the current OTD based FWI methods use acoustic wave equation to simulate wavefield, and do not consider attenuation when inverting velocity. Nevertheless, the real medium usually has a strong viscous property, and seismic data presents amplitude attenuation and phase distortion along time, offset and shot directions. In this case, the current low-dimensional OTD based FWI is not applicable any more. To resolve these two problems, this project studies high-dimensional OTD objective function based viscoacoustic FWI, and takes attenuation into account. The high-dimensional OTD objective function helps to avoid the cycle-skipping problem, thus enhance the convergence of FWI. The new objective function can also take full advantage of attenuation variation of seismic data along time, offset, and shot directions, which leads to a more accurate velocity inversion result.

全波形反演（FWI）是当前众多速度建模方法之中精度最高的，但当地震数据缺乏低频成分，而初始模型的背景信息又不准确时，FWI因存在周波跳跃问题而不收敛。最优传输距离目标函数具有很好的凸性，可避免FWI的周波跳跃，但基于该目标函数的FWI还存在许多不足。首先，最优传输距离目标函数的优势在于计算高维数据的整体差异，而现有方法只在局部维度上（逐道或逐炮）计算最优传输距离，未充分发挥最优传输距离目标函数的优势。其次，现有方法未考虑介质的粘滞性，当地层为衰减介质时，地震数据的振幅和相位沿时间、偏移距和炮线三个方向都有明显的变化，现有的低维度最优传输距离目标函数并不适用于衰减介质。在此背景下，本项目研究基于高维最优传输距离目标函数的粘滞声波FWI，利用最优传输距离目标函数①抑制FWI的周波跳跃问题，增强收敛性；②充分考虑沿时间、偏移距和炮点方向包含的地震波衰减信息，提高FWI的反演精度。

全波形反演是当前最热门的地震成像技术之一，也是被寄予厚望的工业界下一代高精度地震波速度建模方法。然而，全波形反演是高度非线性的反问题，基于传统L2范目标函数的全波形反演多解性严重，反演效果对初始模型和数据低频有很强的依赖性。此外，现有的全波形反演方法大多基于地层完全弹性假设，未考虑介质的粘滞性，导致速度反演精度低。为此，本项目研究适用于衰减介质的全波形反演速度建模方法，具体包括粘滞波动方程正演模拟方法、高维地震数据的最优传输距离目标函数构建方法、以及基于最优传输距离目标函数的全波形反演方法三个方面。总的研究目标是在考虑介质粘滞性的情况下，利用高维最优传输距离目标函数缓解全波形反演的周波跳跃问题，形成稳健、高效的速度反演方法。所取得的具体研究成果包括：一种适用于分数阶拉普拉斯算子粘滞声波方程的矩阵转换数值解法，该方法避免了传统伪谱法因傅里叶变换而引入的周期边界，能更灵活地模拟包括自由边界在内的各类边界条件；一种适用于衰减介质的高效震源波场逆时重构方法，提高了全波形反演的计算效率；一种基于一维Wasserstein-2（W2）和二维Wasserstein-1（W1）最优传输距离目标函数的全波形反演方法，该类目标函数的稳健性明显优于传统L2范数目标函数，能有效缓解全波形反演对低频的依赖，克服周波跳跃问题；一种不依赖于震源子波的卷积型最优传输距离目标函数及全波形反演方法，拓宽了最优传输距离目标函数的适用性；一种稳定的梯度衰减补偿预处理方法，能改善深层反演效果。本项目在全波形反演速度建模方面所取得的理论成果有助于全波形反演这一重要变革性技术的工业化应用，能为解决油气地震勘探领域高精度速度建模问题提供强有力的技术手段。