A CRLB-based statistical framework for interpreting diffraction-limited localization in Lamb wave array imaging

Ultrasonic phased array imaging based on Lamb waves offers accurate defect localization in non-destructive testing, where multimodal dispersion improves responsiveness to structural anomalies. As the classical technique, the delay-and-sum (DAS) beamforming algorithm is distinguished by its computational efficiency and robustness to noise; it is further improved by the total focusing method (TFM) which synthetically focuses at every point in the imaging region, thereby maximizing spatial resolution and defect detectability across the entire field. However, the resolution of TFM imaging, as with any other array imaging, is fundamentally constrained by the diffraction limit, prescribed by Rayleigh’s criterion. Moreover, its effectiveness may be obscured by the multi-mode propagation of Lamb waves which introduces mode interference and overlapping wavefronts. Seeking to understand and quantify the fundamental factors governing imaging resolution and the inherent limitations in Lamb wave-based beamforming detection is therefore crucial to enhance TFM defect localization performance. In this study, we derive the Cramér–Rao lower bound (CRLB) for defect localization under a classical envelope-based, group-delay Lamb-wave DAS/TFM model. The resulting bound is interpreted as a statistical, model-conditioned benchmark for localization uncertainty of guided-wave imaging. Notably, under the adopted imaging formulation, the CRLB is found to follow the same principal wavelength-aperture trends with diffraction-limited point spread function (PSF) broadening, thereby providing an interpretable basis for comparing the localization behaviors of multi-modal Lamb wave. Extensive numerical simulations and experimental study are conducted; the results are observed to be consistent with these theoretical derivations. Particularly, the effect of different modes, including multi-modes, on the imaging resolution is studied; it is found that the introduction of the A1 mode of Lamb waves provides the most substantial improvement in resolution, possibly because the A1 mode exhibits higher group velocity, combined with its steeper dispersion curve slope, thereby enabling more precise and clearer resolution in detecting defects. Overall, this work contributes to a novel rigorous quantitative framework for understanding the resolution constraints in Lamb wave-based TFM and potentially in broader array imaging. The applicability of this work and further study needed in future research are also discussed.