Conventional methods solve the full-waveform inversion making use of gradient-based algorithms to minimize an error function, which commonly measure the Euclidean distance between observed and predicted waveforms. This deterministic approach only provides a ‘best-fitting’ model and cannot account for the uncertainties affecting the predicted solution. Local methods are also usually prone to get trapped into local minima of the error function. On the other hand, casting this inverse problem into a probabilistic framework has to deal with the formidable computational effort of the Bayesian approach when applied to non-linear problems with expensive forward evaluations and large model spaces. We present a gradient-based Markov Chain Monte Carlo full-waveform inversion in which the posterior sampling is accelerated by compressing the data and model spaces through the discrete cosine transform, and by also defining a proposal that is a local, Gaussian approximation of the target posterior probability density. This proposal is constructed using the local Hessian and gradient informations of the log posterior, which are made computationally manageable thanks to the compression of the data and model spaces. We demonstrate the applicability of the approach by performing two synthetic inversion tests on portions of the Marmousi and BP acoustic model. In these examples, the forward modelling is performed using Devito, a finite difference domain-specific language that solves the discretized wave equation on a Cartesian grid. For both examples, the results obtained by the implemented method are also validated against those obtained using a classic deterministic approach. Our tests illustrate the efficiency of the proposed probabilistic method, which seems quite robust against cycle-skipping issues and also characterized by a computational cost comparable to that of the local inversion. The outcomes of the proposed probabilistic inversion can also play the role of starting models for a subsequent local inversion step aimed at improving the spatial resolution of the probabilistic result, which was limited by the model compression.

中文翻译：

---

计算高效的全波形反演贝叶斯方法

传统方法利用基于梯度的算法来最小化误差函数来解决全波形反演，该算法通常测量观测波形和预测波形之间的欧几里德距离。这种确定性方法仅提供“最佳拟合”模型，无法解释影响预测解决方案的不确定性。局部方法通常也容易陷入误差函数的局部最小值。另一方面，当贝叶斯方法应用于具有昂贵的前向评估和大模型空间的非线性问题时，将这种逆问题转化为概率框架必须处理贝叶斯方法的巨大计算量。我们提出了一种基于梯度的马尔可夫链蒙特卡洛全波形反演，其中通过离散余弦变换压缩数据和模型空间，并定义一个目标的局部高斯近似提案来加速后采样后验概率密度。该提案是使用对数后验的局部 Hessian 和梯度信息构建的，由于数据和模型空间的压缩，这些信息在计算上是可管理的。我们通过对 Marmousi 和 BP 声学模型的部分部分进行两次综合反演测试来证明该方法的适用性。在这些示例中，正演建模是使用 Devito 执行的，Devito 是一种有限差分领域特定语言，可求解笛卡尔网格上的离散波动方程。对于这两个示例，所实现的方法获得的结果也根据使用经典确定性方法获得的结果进行了验证。我们的测试说明了所提出的概率方法的效率，该方法对于循环跳跃问题似乎非常稳健，并且其计算成本与局部反演的计算成本相当。所提出的概率反演的结果还可以起到后续局部反演步骤的启动模型的作用，该步骤旨在提高概率结果的空间分辨率，这受到模型压缩的限制。