# Improved Defect Detection Using Adaptive Leaky NLMS Filter in Guided-Wave Testing of Pipelines
## 2 Background Information  
Guided waves are multimodal [26]. Depending on the excitation frequency, multiple wave modes can be generated at the point of excitation. These modes are categorized based on their displacement patterns (mode shapes) within the structure. The main categories of guided waves in pipes are axisymmetric torsional and longitudinal waves and non-axisymmetric flexural waves. The popular nomenclature used for them is in the format of \(X\)(\(n\),\(m\)), where \(X\) can be replaced by letters \(L\) for longitudinal, \(T\) for torsional, and \(F\) for flexural waves; \(n\) shows the harmonic variations of displacement and stress around the circumference; and \(m\) represents the order of existence of the wave mode [27]. In an ideal scenario, a single pure family of axisymmetric waves in their non-dispersive region should be excited for general inspection. Nonetheless, this is not possible due to the variations of excitation and the reception transfer function of transducers in real-life scenarios. Furthermore, the interaction of wave modes with the pipe also leads to the generation of different wave modes, which is commonly known as a wave mode conversion phenomenon [26; 28; 29]. Figure 1 shows the spatial arrival of the signals from different segments of the pipe. These signals are created using the developed model in [30], where a three ring excitation system is used. Each ring contains 32 source points with variable transfer functions. In the figures, blue and red lines show the received amplitude from the transducers of two separate rings while the black dotted line serves as  
the reference. Any point outside the circle illustrates positive amplitude and the points inside show the negative amplitude of the corresponding source point. The following points can be concluded by comparing the illustrated polar plots:  
* The energy of waves received from axisymmetric features (e.g., pipe end) are overwhelmed by the T(0,1) with little variance caused by the existing coherent noise of flexural waves. * Defect signals have a mixture of both flexural and torsional wave modes; nonetheless, if the cross-sectional area (CSA) of the defect is wide, the T(0,1) will have a greater effect than the flexurals. This is the main reason why adding all the transducers tends to increase the overall energy of T(0,1) while reducing the effect of flexurals. * Noise regions are mostly consistent with flexurals with almost no axisymmetric wave being detected. Furthermore, guided waves exhibit dispersive propagation. Dispersion causes the energy of a signal to spread out in space and time as it propagates [32]. Figure 2a shows an example of a dispersion curve calculated from an 8-inch schedule 40 steel pipe using RAPID software [33]. [PERSON] et al. introduced a method to use dispersion curves in order to both simulate [34] and remove [32] the effect of dispersion. By applying the developed equation by [PERSON] et al. and the shown dispersion curves in Figure 2a, a simulated propagation of F(4,2) wave mode is created and shown in Figure 2b where the black and red lines show the same wave mode after 1 m and 2 m of propagation, respectively. As can be seen, for a dispersive waveform the temporal resolution and the energy of wave packet is decreased. Therefore, in general inspection, non-dispersive regions of wave modes are excited to allow ease of inspection. In this research, T(0,1) is used as the excitation wave mode since it is non-dispersive across its whole frequency range [35]. Figure 1: Example of received FEM signals from the array from different sections of a pipe [31]. Figure 2: (a) Example dispersion curve of T(0,1) wave mode in an 8-inch schedule 40 steel pipe. (b) The effect of dispersion on a simulated flexural wave, two propagation distances [31]. Three main signals are expected to be received in guided-wave tests which are shown in Figure 3. The first signal is the superposition of T(0,1) and flexurals where it is the true defect location.