Variational Bayesian Inversion Approaches for Ultrasonic Tomography

超声波无损评估（NDE）对于英国老化工业基础设施的结构评估以及现代增材制造方法的监测和质量控制至关重要。超声无损检测领域的传统成像算法通常假设被检测的部件主要是均匀的，而非均匀性发生在亚波长尺度上。不幸的是，在许多感兴趣的材料中，例如碳纤维增强聚合物（CFRPs）或多晶焊缝，这种假设是无效的，并且可以显着降低检测嵌入缺陷的概率。然而，对组件内空间变化的材料特性的先验知识可以允许纠正时域成像算法通常基于的延迟规律，并导致更可靠的缺陷检测和成像。超声走时层析成像提出了一种切实可行的非破坏性的方法来恢复空间依赖的材料性质的组件从测量其表面。由于层析反演具有明显的非线性，蒙特卡罗（MC）采样方法通常用于此目的，但对于大型数据集和高维参数空间，它们通常在计算上难以处理。这个项目的目的是研究变分贝叶斯反演（VBI）方法取代MC方法的潜力。这些方法将贝叶斯反演问题表述为确定性优化问题，通过最小化Kullback-Leibler （KL）散度，从预定义的概率分布族中寻求后验分布的近似值。通过这种方式，可以以更低的计算成本获得估计后验分布的封闭形式表达式（与使用MC方法获得的数值近似相反）。本项目将把这种方法应用于两个截然不同且与工业相关的逆问题：a)通过在部件边界上进行的超声波走时测量重建速度场。b)构件边界超声走时测量重建局部各向异性刚度图。然后，组件的空间变化材料属性的重建图将与现有的成像算法结合使用，以量化在缺陷检测能力方面取得的改进。

Ultrasonic non-destructive evaluation (NDE) is critical for the structural assessment of the UK's aging industrial infrastructure, as well as for the monitoring and quality control of modern additive manufacturing methods. Traditional imaging algorithms within the ultrasonic NDE community typically assume that the component being inspected is primarily homogeneous, with heterogeneities occurring at sub-wavelength scales. Unfortunately, in many materials of interest, for example carbon fibre reinforced polymers (CFRPs) or polycrystalline welds, this assumption is invalid and can significantly lower the probability of detection of embedded defects. However, prior knowledge of the spatially varying material properties within the component can allow correction of the delay laws on which time domain imaging algorithms are usually based and result in more reliable detection and imaging of defects. Ultrasonic travel time tomography presents a practicable non-destructive approach to recovering the spatially dependent material properties of a component from measurements taken on its surface. Since tomographic inversion is significantly nonlinear, Monte Carlo (MC) sampling methods are often used for this purpose, but they are generally computationally intractable for large datasets and high-dimensional parameter spaces. The aim of this project is to examine the potential of Variational Bayesian Inversion (VBI) methods in place of MC approaches. These methods formulate the Bayesian inversion problem as a deterministic optimisation problem, seeking an approximation to the posterior distribution from a pre-defined family of probability distributions through the minimisation of the Kullback-Leibler (KL) divergence. In this way, a closed form expression estimating the posterior distribution can be achieved (in contrast to the numerical approximation obtained using MC methods) at much lower computational expense. This project will apply this approach to two distinct and industrially relevant inverse problems:a) The reconstruction of velocity fields from ultrasonic travel time measurements made on the boundary of the component.b) The reconstruction of locally anisotropic stiffness maps from ultrasonic travel time measurements made on the boundary of the component.The reconstructed maps of the component's spatially varying material properties will then be used in conjunction with existing imaging algorithms to quantify the improvement achieved in flaw detection capabilities.