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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Fig. 1. (a) Back-propagated baseline signals using the time shift based on the calibrated group velocity ; (b) Back-propagated residual signals using the time shift based on the calibrated group velocity.

if the pixel ( x , y ) is the true location of the damage, this back-propagation will cause the scattered signals from all transmitter-receiver pairs to align constructively, as illustrated by the alignment of their envelopes in Figure 1(b). A set of back-propagated residual signals is obtained for a location ( x , y ) can be stored in a M × N matrix (this matrix will be used in MVDR and MUSIC algorithms below): U xy = ⃗ u xy ( t 1 ) , . . .⃗ u xy ( t n ) , . . .⃗ u xy ( t N ) (10) and⃗ u xy ( t ) = ˆ u residual 1 t + d 1 xy c g , . . . ˆ u residual m t + d mxy c g , . . . ˆ u residual M t + d Mxy c g T (11) where d mxy is the total propagation distance from the transmitter to a pixel at ( x , y ) and then to the receiver for the m-th transmitter-receiver pair. The term ˆ u residual m ( t ) represents the envelope of the m-th residual signal, and c g is the group velocity found during the calibration. t n , n ∈ 1 , 2 , . . . , N in Eq.(10) denotes discrete signal sampling times. 2.2.3. Pixel value calculation Once the back-propagated signals are computed for each pixel (x,y), the Delay-and-Sum imaging value is found by summing these signals across all channels and integrating the result over the excitation time window.This is equivalent to extracting the columns for the back-propagated signal matrix U xy that fall within the excitation window, and then summing all elements in the resulting sub-matrix::

excitation window ˆ u

residual m t +

c g

M m = 1 t ∈

M m = 1 t n ∈

d mxy

U xy ( m , t n )

dt =

P x , y =

(12)

excitation window