84
L. L. Ting et al.
Fig. 8.4 Normal distribution curve with mean of 1
Hence, the obtained localization results were tested whether they are normally
distributed. The portion of results falls within a certain number of σ from μ is
calculated by using Eq. (8.4).
8.2.4 Integration of DTCWT and Soft Threshold
In physics, wave velocity is dispersive because it varies with transmitting medium,
surrounding media, pipe properties and wave frequency [22]. Therefore, the superiority of employing multilevel DTCWT decomposition before localize leak is that
the signal will be decomposed into several frequency bands in which their frequency
range at each level can be calculated as follow:
f (d n ) ∈
f s
2 n+1 ,
f s
2 n
f (a n ) ∈
0,
f s
2 n+1
(8.5)
where d n is detail coefficients, a n is approximation coefficients and n is level.
In such a case, the dispersion phenomenon is taken into account because the
velocity of a wave is evaluated based on its frequency. Figure 8.5 is a dispersion
curve plotted using PCDISP software [23]. There are three elemental modes which
are longitudinal, flexural and torsional modes. Since the sensors placed on the pipe in
a radial direction, the sensors are sensitive to the waves which have significant radial
displacement. Torsional wave only contributes circumferential mode, θ displacement
L. L. Ting et al.
Fig. 8.4 Normal distribution curve with mean of 1
Hence, the obtained localization results were tested whether they are normally
distributed. The portion of results falls within a certain number of σ from μ is
calculated by using Eq. (8.4).
8.2.4 Integration of DTCWT and Soft Threshold
In physics, wave velocity is dispersive because it varies with transmitting medium,
surrounding media, pipe properties and wave frequency [22]. Therefore, the superiority of employing multilevel DTCWT decomposition before localize leak is that
the signal will be decomposed into several frequency bands in which their frequency
range at each level can be calculated as follow:
f (d n ) ∈
f s
2 n+1 ,
f s
2 n
f (a n ) ∈
0,
f s
2 n+1
(8.5)
where d n is detail coefficients, a n is approximation coefficients and n is level.
In such a case, the dispersion phenomenon is taken into account because the
velocity of a wave is evaluated based on its frequency. Figure 8.5 is a dispersion
curve plotted using PCDISP software [23]. There are three elemental modes which
are longitudinal, flexural and torsional modes. Since the sensors placed on the pipe in
a radial direction, the sensors are sensitive to the waves which have significant radial
displacement. Torsional wave only contributes circumferential mode, θ displacement