Detection of abrupt spatial changes in physical properties representing unique geometric features such as buried objects, cavities, and fractures is an important problem in geophysics and many engineering disciplines. In this context, simultaneous spatial field and geometry estimation methods that explicitly parameterize the background spatial field and the geometry of the embedded anomalies are of great interest. This paper introduces an advanced inversion procedure for simultaneous estimation using the domain independence property of the Karhunen-Lo\`eve (K-L) expansion. Previous methods pursuing this strategy face significant computational challenges. The associated integral eigenvalue problem (IEVP) needs to be solved repeatedly on evolving domains, and the shape derivatives in gradient-based algorithms require costly computations of the Moore-Penrose inverse. Leveraging the domain independence property of the K-L expansion, the proposed method avoids both of these bottlenecks, and the IEVP is solved only once on a fixed bounding domain. Comparative studies demonstrate that our approach yields two orders of magnitude improvement in K-L expansion gradient computation time. Inversion studies on one-dimensional and two-dimensional seepage flow problems highlight the benefits of incorporating geometry parameters along with spatial field parameters. The proposed method captures abrupt changes in hydraulic conductivity with a lower number of parameters and provides accurate estimates of boundary and spatial-field uncertainties, outperforming spatial-field-only estimation methods.

翻译：检测代表独特几何特征（如埋藏物体、空腔和裂缝）的物理性质中的空间突变，是地球物理学和众多工程领域中的一个重要问题。在此背景下，能够显式参数化背景空间场与嵌入异常体几何形状的同步空间场与几何估计方法备受关注。本文利用Karhunen-Loève（K-L）展开的域独立性特性，提出了一种用于同步估计的先进反演流程。以往遵循此策略的方法面临显著的计算挑战：相关的积分特征值问题需要在不断演化的域上重复求解，而基于梯度的算法中的形状导数需要计算代价高昂的Moore-Penrose逆。利用K-L展开的域独立性，所提方法避免了这两个瓶颈，积分特征值问题仅需在固定的边界域上求解一次。对比研究表明，我们的方法在K-L展开梯度计算时间上实现了两个数量级的提升。在一维和二维渗流问题上的反演研究突显了将几何参数与空间场参数共同纳入的益处。所提方法能够以更少的参数捕捉水力传导率的突变，并提供边界及空间场不确定性的精确估计，其性能优于仅估计空间场的方法。