SIAM Journal on Imaging Sciences, Volume 14, Issue 1, Page 58-91, January 2021.
Full-waveform inversion (FWI) is a nonlinear PDE constrained optimization problem which seeks to estimate the constitutive parameters of a medium by fitting waveforms. Among these parameters, attenuation needs to be taken into account in viscous media to exploit the full potential of FWI. Attenuation is easily implemented in the frequency domain by using complex-valued velocities in the time-harmonic wave equation. These complex velocities are frequency-dependent to guarantee causality and account for dispersion. Since estimating a complex frequency-dependent velocity at each grid point in space is not realistic, the optimization is generally performed in the real domain by processing the phase velocity (or slowness) at a reference frequency and attenuation (or quality factor) as separate real parameters. This real parametrization requires an a priori empirical relation (such as the nonlinear Kolsky--Futterman (KF) or standard linear solid (SLS) attenuation models) between the complex velocity and the two real quantities, which is prone to generate modeling errors if it does not represent accurately the attenuation behavior of the medium. Moreover, it leads to a multivariate inverse problem, which is ill-posed due to the cross-talk between the two classes of real parameters. To alleviate these issues, we solve directly the optimization problem in the complex domain by processing narrow bands of frequencies in sequence under the assumption of bandwise frequency dependence of the complex velocities. Moreover, we use a relaxation method to extend the FWI search space by processing the wave equation as a weak constraint with the alternating direction method of multipliers (ADMM) to mitigate the risk of spurious local minima. To mitigate the ill-posedness of the inversion, three total variation (TV) regularization schemes based upon ADMM and proximity algorithms are presented. In the first, regularization is applied directly on the complex velocities. In the other two, separate TV regularizations are tailored to different attributes of the complex velocities (real and imaginary parts, magnitude and phase). The real phase velocity and attenuation factor are then reconstructed a posteriori at each spatial position from the estimated complex velocity using arbitrary empirical relation. The numerical results first show that the regularization of the amplitude and phase provides the most reliable results. Moreover, they show that the band-by-band design of the inversion limits the sensitivity of the recovered phase velocity and attenuation factor to the attenuation model used for their a posteriori extraction.

中文翻译：

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具有总变化正则化的复值成像：粘声介质中全波形反演的应用

SIAM影像科学杂志，第14卷，第1期，第58-91页，2021年1月。
全波形反演（FWI）是非线性PDE约束优化问题，旨在通过拟合波形来估计介质的本构参数。在这些参数中，需要在粘性介质中考虑衰减，以充分利用FWI的潜力。通过在时谐波方程中使用复数值速度，可以轻松地在频域中实现衰减。这些复杂的速度取决于频率，以确保因果关系并解决色散。由于估算空间中每个网格点处与频率相关的复杂速度是不现实的，因此优化通常是在实域中通过将参考频率下的相速度（或慢度）和衰减（或品质因数）作为单独的实数进行处理而进行的。参数。这种真实的参数化要求复数速度和两个实数之间存在先验经验关系（例如非线性Kolsky-Futterman（KF）或标准线性固体（SLS）衰减模型），如果它会容易产生建模误差不能准确表示介质的衰减行为。而且，这导致了多元逆问题，由于两类实参之间的串扰，该问题不适当。为了缓解这些问题，我们在复杂速度的频带间频率依赖性的假设下，通过顺序处理窄频带来直接解决复杂域中的优化问题。而且，我们使用松弛法通过将波动方程视为弱约束，并使用乘数交变方向法（ADMM）将其作为弱约束来扩展FWI搜索空间，以减少伪局部极小值的风险。为了减轻反演的不适定性，提出了三种基于ADMM和邻近算法的总变分（TV）正则化方案。首先，将正则化直接应用于复数速度。在另外两个中，针对复杂速度（实部和虚部，幅度和相位）的不同属性量身定制了单独的电视正则化。然后，使用任意经验关系从估计的复数速度在每个空间位置重建后验真实速度和衰减因子。数值结果首先表明，振幅和相位的正则化提供了最可靠的结果。此外，他们表明，反演的逐个频带设计将恢复的相速度和衰减因子的灵敏度限制在用于其后验提取的衰减模型上。