Cramer-Rao Lower Bound for Performance Analysis of Leak Detection

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.