PSI - Issue 80

Guangxiao Zou et al. / Procedia Structural Integrity 80 (2026) 93–104 Author name / Structural Integrity Procedia 00 (2023) 000–000

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noise w measured − baseline [ t ]:

u measured − baseline [ t ]

direct [ t ]

boundary − reflection [ t ]

measured − baseline [ t ]

= u

+ u

+ w

(7)

Assuming the direct wave and boundary reflection waves are the same for both the damaged and intact states , the residual signal obtained by the baseline subtraction will be:

u residual [ t ]

= u measured − damage [ t ] − u measured − baseline [ t ]

scattered [ t ]

= u

+ w [ t ]

(8)

Therefore, the residual signal is approximately equal to the damage-scattered signal, corrupted by a noise term, w ( t ), which represents the di ff erence between the noise in the measurements from the damaged and intact states [Eqs. (6) and (7)]. However, in practice, residual noise and boundary reflection di ff erence between damaged and intact states degrade imaging performance, leading to poor resolution and the formation of artifacts.

2.2. DAS imaging

2.2.1. Calibration of group velocity An accurate group velocity is crucial for all of the imaging algorithms that are considered in this paper. A calibration is performed using baseline signals recorded from the undamaged plate to obtain the group velocity. The method works by computationally time-shifting or ’back-propagating’ each recorded signal’s first wave packet to align them within a specific time window that corresponds to the duration of the excitation signal Hall et al. (2011). This time shift is calculated by dividing the known distance between the transmitter and receiver by a candidate group velocity. This process is repeated for a range of possible velocities. To avoid errors caused by the signal changing shape due to dispersion, the envelope of the signal is used instead of the raw waveform. The optimal group velocity c g is the one that causes the back-propagated signal envelopes from all transmitter-receiver pairs to align more closely, which is a maximization of the following expression:

M m = 1 ˆ u

t +

c dt

c t ∈ excitation window

d m

measured − baseline m

c g = argmax

(9)

In this formula, M is the total number of unique transmitter-receiver pairs. The term ˆ u measured − baseline m represents the complex envelope of the m-th filtered baseline signal, and d m is the distance for that specific pair. The signal’s real valued envelope is found by taking the absolute value of this complex envelope. The integration period is set to 0-100 µ s to match the duration of the excitation signal. The successful alignment of the back-propagated baseline envelopes, shown in Figure 1(a), validates the accuracy of the group velocity obtained through calibration. Furthermore, as shown in Figure 1(b), applying this calibrated velocity to the damage-scattered signals results in their successful alignment within the excitation window. 2.2.2. Back-propagated residual signals The imaging algorithm systematically tests every potential damage location, or pixel ( x , y ), within the region of interest. For each pixel, the total propagation distance from a transmitter to ( x , y ) and then to a receiver is calculated. This distance, divided by the previously calibrated group velocity, gives the expected travel time for that specific path. This travel time is then used to time-shift, or ”back-propagate,” the corresponding residual signal (as established in Eq. (8), the residual signal is a close approximation of the actual damage-scattered signal). The core principle is that