## 1 Introduction
[22] investigated the scattering patterns of interaction between S0 Lamb waves and fatigue cracks from analytical, numerical, and experimental perspectives. They found that an embryonic fatigue crack in an aluminum plate waveguide can be oriented accurately. [PERSON] et al. [23] investigated the influence of the length of the fatigue crack as well as the incident wave angle on the amplitude of second harmonic generated by S0 mode. Although researchers have studied the interaction between nonlinear Lamb waves and interface nonlinearity, few studies have been reported on localization and imaging of micro-damage using nonlinear Lamb waves [24; 25; 26; 27]. In the application of Lamb wave for SHM, appropriate damage identification algorithms have usually been adopted to achieve the damage position by giving an interpretable and intuitive image in a three-dimensional structure. One of the most common damage imaging algorithms is based on the so-called ellipse method [28; 29], which determines the damage localization by the intersection of the elliptical loci based on the time of flight (ToF) and assumes that the group velocity remains unchanged before and after interacting with the damage. This will inevitably cause positioning errors due to the change of the group velocity before and after interaction with the damage [30]. When using the mode pair S0-s0 with quasi group-velocity matching, other candidate positioning methods will be implemented for damage localization. A total focus method based on decomposition of the time reversal operator was experimentally investigated by [PERSON] et al. [24] to locate two fatigue cracks in an aluminum plate. A probabilistic algorithm based on probability scan matrix was also implemented by [PERSON] et al. [25] to localize a microcrack in a thin plate, but it was considered that the microcrack had the highest probability of occurrence on the propagation path with a much dense transducer configuration. Furthermore, a damage is generally not a single point, but with a certain dimension [31]. [PERSON] et al. [26] proposed a probability-based diagnostic imaging (PDI) based on elliptical loci to determine the most possible intersection areas for damage localization. It was demonstrated that the method of probability distributions can be used to predict or estimate damage area. [PERSON] et al. [27]localized a fatigue crack starting at a rivet hole in an aluminum plate by a probabilistic imaging algorithm using nonlinear Lamb waves, in which the employed mode pair S1-s2 needed to exactly meet the phase- and group-velocity matching. Therefore, the objective of this study is to propose a modified ellipse method for accurate localization and imaging of a micro-crack in a three-dimensional structure using nonlinear Lamb waves with imperfect group-velocity matching. This study is organized as follows. Section 2 introduces the concept of the modified method and establishes the mathematical formulation for damage localization using nonlinear S0 mode Lamb waves with quasi group-velocity matching. Section 3 provides the details of the experimental setup to estimate the method's performance for localization of a small-crack in an aluminum plate. The experimental results for localization of the small-crack based on linear and nonlinear features are discussed, and the identification results of the proposed method were compared with that of the traditional ellipse method. Section 4 presents finite element simulations of a breathing crack in comparison with the experimental verification. The proposed method is also validated when the breathing crack is not situated in the propagation path. Finally, conclusions are drawn in the Section 5.