Detection of Background Water Leaks Using a High-Resolution Dyadic Transform

Abstract

This article solves the problem of detecting water leaks with a minimum size of down to 1 mm in diameter. Two new mathematical tools are used to solve this problem: the first one is the Te cross-spectral density and the second is Te coherence. These mathematical tools provide the possibility of discriminating spurious frequency components, making use of the property of multi-sensitivity. This advantage makes it possible to maximize the sensitivity of the frequency spectrum. The wavelet function used was Daubechies 45, because it provides an attenuation of 150 dB in the rejection band. The tools were validated with two scenarios. For the first scenario, a synthetic signal was analyzed. In the second scenario, two types of background leakage were analyzed: the first one has a diameter of 1 mm with a signal-to-noise ratio of 2.82 dB and flow rate of 33.7 mL/s, and the second one has a diameter of 4 mm with a signal-to-noise ratio of 9.73 dB with a flow rate of 125.0 mL/s. The results reported in this paper show that both the Te cross-spectral density and Te coherence are higher than those reported in scientific literature.

The Te Transform: A High-Resolution Integral Transform and Its Key Properties

“…Furthermore, the fifth contribution demonstrates the existence of the inverse T e transform, and lastly, the sixth contribution provides a uniqueness theorem for the T e transform. The theoretical results presented in this paper were validated by the authors of the works shown in [1][2][3][4].…”

“…Only when compacting its output information is it observed how the frequency is scaled by a dyadic factor. Previous research conducted by the authors of this paper demonstrates that such scaling enables the isolation of frequency components, the attenuation of cross-terms, and the identification of system behaviors [1][2][3][4].…”

“…Recently, the authors of [3,4] have utilized the T e transform as a basis in hydraulic and seismic applications. They reported as the main innovation the T e cross-spectral density, along with T e coherence and the T e gram, respectively.…”

Feature Extraction of a Non-Stationary Seismic–Acoustic Signal Using a High-Resolution Dyadic Spectrogram

“…For the T e -gram definition in Section 3, it is necessary to start from the definition of the T e Transform given by the authors in [4,32]. It is also required for the analysis of the seismic signal that will be developed in Section 5.…”

“…The T e Transform defined by [4,32] is a mathematical tool that allows for obtaining a dyadic frequency spectrum. Its main advantage is that it allows the isolation of spurious frequency components.…”