Reconstructing the structure of the soil using non-invasive techniques is a very relevant problem in many scientific fields, like geophysics and archaeology. This can be done, for instance, with the aid of Frequency Domain Electromagnetic (FDEM) induction devices. Inverting FDEM data is a very challenging inverse problem, as the problem is extremely ill-posed, i.e., sensible to the presence of noise in the measured data, and non-linear. Regularization methods substitute the original ill-posed problem with a well-posed one whose solution is an accurate approximation of the desired one. In this paper we develop a regularization method to invert FDEM data. We propose to determine the electrical conductivity of the ground by solving a variational problem. The minimized functional is made up by the sum of two term: the data fitting term ensures that the recovered solution fits the measured data, while the regularization term enforces sparsity on the Laplacian of the solution. The trade-off between the two terms is determined by the regularization parameter. This is achieved by minimizing an $\ell_2 - \ell_q$ functional with $0 < q \leq 2$. Since the functional we wish to minimize is non-convex, we show that the variational problem admits a solution. Moreover, we prove that, if the regularization parameter is tuned accordingly to the amount of noise present in the data, this model induces a regularization method. Some selected numerical examples on synthetic and real data show the good performances of our proposal.
翻译:使用非侵入技术重建土壤结构在许多科学领域,例如地球物理学和考古学,都是一个非常相关的问题。例如,借助频度 Domain 电磁感应装置(FDEM) 的帮助,可以做到这一点。反转 FDEM 数据是一个极具挑战性的反向问题,因为这个问题极不可靠,即对测量数据中的噪音和非线性数据而言是明智的。常规化方法取代了最初的不成熟的合成问题,而其解决方案准确接近所希望的。在本文中,我们开发了一种正规化方法,用于倒转 FDEM 数据。我们建议通过解决变异问题来确定地面的电导率。最弱的功能由两个术语的总和组成:数据正确性术语确保回收的解决方案符合测量数据,而常规化术语则强化了解决方案的模型。两种术语之间的交易由常规化参数决定。在正度参数上,我们通过最小化一个 $\\ 值的功能性能变化数据显示一个不是正常的数值。