Cramer-Rao Lower Bound for Performance Analysis of Leak Detection

Abstract

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.

Experimental investigation of transients-induced fluid–structure interaction in a pipeline with multiple-axial supports

“…Reservoir head (m) the experiment should be conducted such that the generated signal contains useful information from the system. For example, in typical transient-based-defect-detection methods, the generated signal should be of high enough frequency bandwidth (thus sharp transients) to provide the desired information from the system (Ferrante and Brunone, 2003;Lee et al, 2013Lee et al, , 2015Gong et al, 2015;Ferrante et al, 2014;Keramat et al, 2018;Keramat and Zanganeh, 2019).…”

Multi‐Sensor Fusion for Transient‐Based Pipeline Leak Localization in the Dempster‐Shafer Evidence Framework

“…where SNR in this equation is in the linear scale. Equation 27shows that multiple sensors provide similar quantity of information about leak (1/I gives the Cramer-Rao lower bound which is the lower limit of leak localization error [Keramat et al, 2019;Wang & Ghidaoui, 2018a] 4, where…”

“…Therefore, the application of the existing methods is often limited. From the perspective of information theory (Keramat et al., 2019; Wang & Ghidaoui, 2018a), there are two ways to increase the robustness of leakage localization. The first is to repeat experiments for multiple times: it is experimentally shown in (Wang, Lin, & Ghidaoui, 2020) that, using the average of seven transient measurements, the wavelet transform method (Ferrante & Brunone, 2003; Ferrante et al., 2007) (a representative of TBRM) is able to accurately localize a small leak where the outflow from the leak is 10% of the main pipe flow.…”

Uncertainty Quantification of Transient‐Based Leakage Identification: A Frequency Domain Framework

“…This estimation is accomplished based on the Fisher information matrix in a frequency domain framework, while Keramat et al. (2018) conducted it on a time‐domain basis. The formulations are verified by the Monte Carlo method applied to the recent localization method developed by the authors (Keramat & Duan, 2021; Keramat et al., 2022).…”

“…This research evaluates the lower bound of the leak parameters covariance matrix, or equivalently, the mean square error of the estimated leak sizes and locations. This estimation is accomplished based on the Fisher information matrix in a frequency domain framework, while Keramat et al (2018) conducted it on a time-domain basis. The formulations are verified by the Monte Carlo method applied to the recent localization method developed by the authors (Keramat & Duan, 2021;Keramat et al, 2022).…”