Cramer-Rao Lower Bound for Performance Analysis of Leak Detection
Publication: Journal of Hydraulic Engineering
Volume 145, Issue 6
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.
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Acknowledgments
This work has been supported by research grants from the Research Grant Council of the Hong Kong SAR, China (Project Nos. T21-602/15R and 16208618). The authors would like to thank Duncan McInnis, Fedi Zouari, Asgar Ahadpour, Man Yue Lam, and Jingrong Lin for their helpful comments and discussions.
References
Ahmadi, A., and A. Keramat. 2010. “Investigation of fluid-structure interaction with various types of junction coupling.” J. Fluids Struct. 26 (7–8): 1123–1141. https://doi.org/10.1016/j.jfluidstructs.2010.08.002.
Bolliger, C. S., C. Boesch, and R. Kreis. 2013. “On the use of Cramér-Rao minimum variance bounds for the design of magnetic resonance spectroscopy experiments.” Neuroimage 83: 1031–1040. https://doi.org/10.1016/j.neuroimage.2013.07.062.
Brunone, B. 1999. “Transient test-based technique for leak detection in outfall pipes.” J. Water Resour. Plann. Manage. 125 (5): 302–306. https://doi.org/10.1061/(ASCE)0733-9496(1999)125:5(302).
Brunone, B., and M. Ferrante. 2001. “Detecting leaks in pressurized pipes by means of transients.” J. Hydraul. Res. 39 (5): 539–547. https://doi.org/10.1080/00221686.2001.9628278.
Chaudhry, M. H. 2014. Applied hydraulic transients. 3rd ed. New York: Springer.
Colombo, A. F., P. Lee, and B. W. Karney. 2009. “A selective literature review of transient-based leak detection methods.” J. Hydro-Environ. Res. 2 (4): 212–227. https://doi.org/10.1016/j.jher.2009.02.003.
Conte, J. P., P. K. Vijalapura, and M. Meghella. 2003. “Consistent finite-element response sensitivity analysis.” J. Eng. Mech. 129 (12): 1380–1393. https://doi.org/10.1061/(ASCE)0733-9399(2003)129:12(1380).
Covas, D., I. Stoianov, J. Mano, H. Ramos, N. Graham, and C. Maksimovic. 2005. “The dynamic effect of pipe-wall viscoelasticity in hydraulic transients. Part II-Model development, calibration and verification.” J. Hydraul. Res. 43 (1): 56–70. https://doi.org/10.1080/00221680509500111.
Diong, M. L., A. Roueff, L. Philippe, and A. Litman. 2015. “Precision analysis based on Cramer-Rao bound for 2D acoustics and electromagnetic inverse scattering.” Inverse Prob. 31 (7): 075003. https://doi.org/10.1088/0266-5611/31/7/075003.
Duan, H. 2016. “Sensitivity analysis of a transient-based frequency domain method for extended blockage detection in water pipeline systems.” J. Water Resour. Plann. Manage. 142 (4): 04015073. https://doi.org/10.1061/(ASCE)WR.1943-5452.0000625.
Ferrante, M., B. Brunone, and S. Meniconi. 2007. “Wavelets for the analysis of transient pressure signals for leak detection.” J. Hydraul. Eng. 133 (11): 1274–1282. https://doi.org/10.1061/(ASCE)0733-9429(2007)133:11(1274).
Ferrante, M., B. Brunone, and S. Meniconi. 2010. “Leak-edge detection.” J. Hydraul. Res. 47 (2): 233–241. https://doi.org/10.3826/jhr.2009.3220.
Ferrante, M., B. Brunone, and S. Meniconi. 2014. “Leak size, detectability and test conditions in pressurized pipe systems.” Water Resour. Manage. 28 (13): 4583–4598. https://doi.org/10.1007/s11269-014-0752-6.
Ferrante, M., C. Capponi, R. Collins, J. Edwards, B. Brunone, and S. Meniconi. 2016. “Numerical transient analysis of random leakage in time and frequency domains.” Civ. Eng. Environ. Syst. 33 (1): 70–84. https://doi.org/10.1080/10286608.2016.1138941.
Ferreira, J. P. B. C. C., N. M. C. Martins, and D. I. C. Covas. 2018. “Ball valve behavior under steady and unsteady conditions.” J. Hydraul. Eng. 144 (4): 04018005. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001434.
Garthwaite, P., I. Jolliffe, and B. Jones. 2002. Statistical inference. 2nd ed. London: Oxford Science Publications.
Greenspan, H. 2008. “Super-resolution in medical imaging.” Comput. J. 52 (1): 43–63. https://doi.org/10.1093/comjnl/bxm075.
Haghighi, A., H. M. Ramos. 2012. “Detection of leakage freshwater and friction factor calibration in drinking networks using central force optimization.” Water Resour. Manage. 26 (8): 2347–2363. https://doi.org/10.1007/s11269-012-0020-6.
Jung, B. S., and B. W. Karney. 2008. “Systematic exploration of pipeline network calibration using transients.” J. Hydraul. Res. 46: 129–137. https://doi.org/10.1080/00221686.2008.9521947.
Kapelan, Z. S., D. A. Savic, and G. A. Walters. 2003. “A hybrid inverse transient model for leakage detection and roughness calibration in pipe networks.” J. Hydraul. Res. 41 (5): 481–492. https://doi.org/10.1080/00221680309499993.
Kay, S. M. 1993. Fundamentals of statistical signal processing. Vol. 1 of estimation theory. Englewood Cliffs, NJ: Prentice-Hall.
Keramat, A., A. Gaffarian Kolahi, and A. Ahmadi. 2013. “Waterhammer modelling of viscoelastic pipes with a time-dependent Poisson’s ratio.” J. Fluids Struct. 43: 164–178. https://doi.org/10.1016/j.jfluidstructs.2013.08.013.
Keramat, A., and A. S. Tijsseling. 2012. “Waterhammer with column separation, fluid-structure interaction and unsteady friction in a viscoelastic pipe.” In Proc., 11th Int. Conf. on Pressure Surges, edited by S. Anderson, 443–460. Bedford, UK: BHR Group.
Keramat, A., A. S. Tijsseling, Q. Hou, and A. Ahmadi. 2012. “Fluid-structure interaction with pipe-wall viscoelasticity during water hammer.” J. Fluids Struct. 28: 434–455. https://doi.org/10.1016/j.jfluidstructs.2011.11.001.
Kleiber, M., H. Antunez, T. D. Hien, and P. Kowalczyk. 1997. Parameter sensitivity in nonlinear mechanics: Theory and finite element computations. Chichester, UK: Wiley.
Lee, P. J., H. F. Duan, M. Ghidaoui, and B. Karney. 2013. “Frequency domain analysis of pipe fluid transient behavior.” J. Hydraul. Res. 51 (6): 609–622. https://doi.org/10.1080/00221686.2013.814597.
Lee, P. J., H. F. Duan, J. Tuck, and M. Ghidaoui. 2015. “Numerical and experimental study on the effect of signal bandwidth on pipe assessment using fluid transients.” J. Hydraul. Eng. 141 (2): 04014074. https://doi.org/10.1061/(ASCE)HY.1943-7900.0000961.
Liggett, J. A., and L. C. Chen. 1994. “Inverse transient analysis in pipe networks.” J. Hydraul. Eng. 120 (8): 934–955. https://doi.org/10.1061/(ASCE)0733-9429(1994)120:8(934).
Louati, M., M. Meniconi, M. S. Ghidaoui, and B. Brunone. 2017. “Experimental study of the eigenfrequency shift mechanism in blocked pipe system.” J. Hydraul. Eng. 143 (10): 04017044. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001347.
Majd, A., A. Ahmadi, and A. Keramat. 2016. “Investigation of non-Newtonian fluid effects during transient flows in a pipeline.” Strojniški vestnik/J. Mech. Eng. 62 (2): 105–115. https://doi.org/10.5545/sv-jme.2015.2787.
McCutchen, C. W. 1967. “Super-resolution in microscopy and the Abbe resolution limit.” J. Opt. Soc. Am. 57 (10): 1190. https://doi.org/10.1364/JOSA.57.001190.
Meniconi, S., B. Brunone, M. Ferrante, and C. Massari. 2014. “Energy dissipation and pressure decay during transients in viscoelastic pipes with an in-line valve.” J. Fluids Struct. 45: 235–249. https://doi.org/10.1016/j.jfluidstructs.2013.12.013.
Meniconi, S., B. Brunone, M. Frisinghelli, E. Mazzetti, M. Larentis, and C. Costisella. 2017. “Safe transients for pipe survey in a real transmission main by means of a portable device: The case study of the Trento (I) supply system.” Procedia Eng. 186: 228–235. https://doi.org/10.1016/j.proeng.2017.03.232.
Nash, G. A., and B. W. Karney. 1999. “Efficient inverse transient analysis in series pipe systems.” J. Hydraul. Eng. 125 (7): 761–764. https://doi.org/10.1061/(ASCE)0733-9429(1999)125:7(761).
Nehorai, A., and M. Hawkes. 2000. “Performance bounds for estimating vector systems.” IEEE Trans. Signal Process. 48 (6): 1737–1749. https://doi.org/10.1109/78.845931.
Oppenheim, A. V., A. S. Willsky, and I. T. Young. 1983. Signals and systems. 15th ed. Englewood Cliffs, NJ: Prentice-Hall.
Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. 1992. Numerical recipes: The art of scientific computing. Cambridge, UK: Cambridge University Press.
Puschmann, K. G., and F. Kneer. 2005. “On super-resolution in astronomical imaging.” Astron. Astrophys. 436 (1): 373–378. https://doi.org/10.1051/0004-6361:20042320.
Robinson, D., and P. Milanfar. 2006. “Statistical performance analysis of super-resolution.” IEEE Trans. Image Process. 15 (6): 1413–1428. https://doi.org/10.1109/TIP.2006.871079.
Sattar, A. M., and M. El-Beltagy. 2017. “Stochastic solution to the water hammer equations using polynomial chaos expansion with random boundary and initial conditions.” J. Hydraul. Eng. 143 (2): 04016078. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001227.
Shannon, C. E. 1949. “Communication in the presence of noise.” Proc. IRE 37 (1): 10–21. https://doi.org/10.1109/JRPROC.1949.232969.
Soares, A. K., D. I. C. Covas, and L. F. Reis. 2008. “Analysis of PVC pipe-wall viscoelasticity during water hammer.” J. Hydraul. Eng. 134 (9): 1389–1394. https://doi.org/10.1061/(ASCE)0733-9429(2008)134:9(1389).
van Trees, H. L., and K. Bell. 2013. Detection, estimation, and modulation theory: I. Detection, estimation, and filtering theory. 2nd ed. New York: Wiley.
Vaseghi, S. 2008. Advanced digital signal processing and noise reduction. 4th ed. Chichester, UK: Wiley.
Vitkovsky, J., M. Lambert, A. Simpson, and J. Liggett. 2007. “Experimental observation and analysis of inverse transients for pipeline leak detection.” J. Water Resour. Plann. Manage. 136 (6): 519–530. https://doi.org/10.1061/(ASCE)0733-9496(2007)133:6(519).
Vitkovsky, J. P., J. Liggett, A. R. Simpson, and M. F. Lambert. 2003. “Optimal measurement site locations for inverse transient analysis in pipe networks.” J. Water Resour. Plann. Manage. 129 (6): 480–492. https://doi.org/10.1061/(ASCE)0733-9496(2003)129:6(480).
Vítkovský, J. P., A. R. Simpson, and M. F. Lambert. 2000. “Leak detection and calibration using transients and genetic algorithms.” J. Water Resour. Plann. Manage. 126 (4): 262–265. https://doi.org/10.1061/(ASCE)0733-9496(2000)126:4(262).
Wang, X., and M. S. Ghidaoui. 2018a. “Identification of multiple leaks in pipeline: Linearized model, maximum likelihood, and super-resolution localization.” Mech. Syst. Signal Process. 107: 529–548. https://doi.org/10.1016/j.ymssp.2018.01.042.
Wang, X., and M. S. Ghidaoui. 2018b. “Pipeline leak detection using the matched-field processing method.” J. Hydraul. Eng. 144 (6): 04018030. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001476.
Wang, X., D. P. Palomar, L. Zhao, M. S. Ghidaoui, and R. D. Murch. 2019. “Spectral-based methods for pipeline leak detection.” J. Hydraul. Eng. 145 (3): 04018089. https://doi.org/10.1061/(ASCE)HY.1943-7900.0001572.
Wylie, E. B., and V. L. Streeter. 1993. Fluid transients in systems. Englewood Cliffs, NJ: Prentice-Hall.
Zhao, M., M. S. Ghidaoui, M. Louati, and H. F. Duan. 2018. “Numerical study of the blockage length effect on the transient wave in pipe flows.” J. Hydraul. Res. 56 (2): 245–255. https://doi.org/10.1080/00221686.2017.1394374.
Zhao, W., J. Yu, A. Zhang, and Y. Li. 2014. “Optimal sensors deployment for tracking level curve based on posterior Cramér-Rao lower bound in scalar field.” In Vol. 8917 of Intelligent robotics and applications: Lecture notes in computer science (including subseries lecture notes in artificial intelligence and lecture notes in bioinformatics), 129–141. Cham, Switzerland: Springer.
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Published In
Journal of Hydraulic Engineering
Volume 145 • Issue 6 • June 2019
Copyright
©2019 American Society of Civil Engineers.
History
Received: Jun 25, 2018
Accepted: Nov 27, 2018
Published online: Mar 29, 2019
Published in print: Jun 1, 2019
Discussion open until: Aug 29, 2019
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