Full waveform inversion (FWI) enables us to obtain high-resolution velocity models of the subsurface. However, estimating the associated uncertainties in the process is not trivial. Commonly, uncertainty estimation is performed within the Bayesian framework through sampling algorithms to estimate the posterior distribution and identify the associated uncertainty. Nevertheless, such an approach has to deal with complex posterior structures (e.g., multimodality), high-dimensional model parameters, and large-scale datasets, which lead to high computational demands and time-consuming procedures. As a result, uncertainty analysis is rarely performed, especially at the industrial scale, and thus, it drives practitioners away from utilizing it for decision-making. This work proposes a frugal approach to estimate uncertainty in FWI through the Stein Variational Gradient Descent (SVGD) algorithm by utilizing a relatively small number of velocity model particles. We warm-start the SVGD algorithm by perturbing the optimized velocity model obtained from a deterministic FWI procedure with random field-based perturbations. Such perturbations cover the scattering (i.e., high wavenumber) and the transmission (i.e., low wavenumber) components of FWI and, thus, represent the uncertainty of the FWI holistically. We demonstrate the proposed approach on the Marmousi model; we have learned that by utilizing a relatively small number of particles, the uncertainty map presents qualitatively reliable information that honours the physics of wave propagation at a reasonable cost, allowing for the potential for industrial-scale applications. Nevertheless, given that uncertainties are underestimated, we must be careful when incorporating them into downstream tasks of seismic-driven geological and reservoir modelling.

翻译：全波形反演（FWI）能够获得地下高分辨率的速度模型，但估算过程中相关的不确定性并非易事。通常，不确定性估算在贝叶斯框架内通过采样算法进行，以估计后验分布并识别相关不确定性。然而，这种方法需要处理复杂的后验结构（如多模态性）、高维模型参数和大规模数据集，导致计算需求高、耗时较长。因此，不确定性分析很少被开展，尤其在工业规模下，从而阻碍了从业者将其用于决策。本文提出一种经济的方法，通过斯坦因变分梯度下降（SVGD）算法，利用相对较少的粒子数量来估算FWI中的不确定性。我们通过随机场扰动对确定性FWI过程优化的速度模型进行热启动，从而初始化SVGD算法。这种扰动覆盖了FWI的散射（高波数）和透射（低波数）分量，从而整体上表征了FWI的不确定性。我们在Marmousi模型上演示了所提出的方法；我们发现，通过使用相对较少的粒子数，不确定性图在合理成本下提供了定性可靠的信息，这些信息尊重了波传播的物理规律，具备工业规模应用的潜力。然而，由于不确定性被低估，在将其纳入地震驱动的地质和储层建模下游任务时需谨慎处理。