Cramer-Rao Lower Bound for Performance Analysis of Leak Detection

Alireza Keramat, Mohamed S. Ghidaoui, Xun Wang, Moez Louati

Research output: Journal article publicationJournal articleAcademic researchpeer-review

22 Citations (Scopus)


Due to random noise in real measurements, leak detection (estimation of leak size and location) is subject to a degree of uncertainty. This paper provides a framework to investigate the lower bound of the variance of a leak's variable estimation and delineates the parameters upon which this lower bound depend. This is accomplished by applying the Cramer-Rao lower bound (CRLB) principle to the leak detection problem. For a given data set, CRLB gives the minimum mean square error of any unbiased estimator. The CRLB is evaluated using the Fisher information, which is evaluated from direct differentiation of the water-hammer characteristics equations. The results show that the CRLB of the leak-size estimate increases with time of closure and noise level but reduces with the duration of the measured signal. It is also shown that the CRLB is instrumental in the systematic design of efficient transient tests for leak detection. The error of leak-size estimates rises remarkably with setting distances between consecutive potential leaks of less than half the minimum wavelength of the probing signal. More conclusions are drawn on appropriate mesh-size for inverse transient analysis (ITA), maximum possible accuracy in successful localization, and its probability subject to the physical situation's parameters.

Original languageEnglish
Article number04019018
JournalJournal of Hydraulic Engineering
Issue number6
Publication statusPublished - 1 Jun 2019
Externally publishedYes


  • Cramer-Rao lower bound
  • Direct differentiation method
  • Inverse transient analysis
  • Leak detection
  • Method of characteristics
  • Noise
  • Performance limits
  • Water hammer

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Water Science and Technology
  • Mechanical Engineering


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