Leak Localization in Looped Pipe Networks Based on a Factorized Transient Wave Model: 2. Regularization of Ill-Conditioned Problems

A previous paper proposed an accurate and efficient matched-field processing method for leak localization in looped pipe networks based on the known discharge and pressure head (i.e., hydraulic properties) at all internal nodes. However, the computation of these internal hydraulic properties from boundary measurements is an ill-conditioned inverse problem: for a small perturbation in the boundary measurements there is a huge change in the internal hydraulic properties. We prove and clarify the invalidity of the classic Tikhonov regularization for this ill-conditioned problem. As an alternative, this paper derives the theoretical upper bounds of noise-induced errors, based on which a threshold is determined for identifying ill-conditioned frequencies. The leak localization is implemented by discarding wave information at all ill-conditioned frequencies. This results in an efficient and robust algorithm for leak localization in noisy environments. Numerical experiments demonstrate that an unknown leak anywhere in looped pipe networks can be accurately localized by the proposed algorithm.