## 1 Introduction  
Lamb waves have shown great potential for structural health monitoring (SHM) in plate-like structures [1]. Their attractive features include sensitivity to a variety of damage types and the capability of traveling relatively long distances [2]. Damage imaging methods based on Lamb waves have been widely studied by many researchers, such as delay-and-sum (DAS) imaging [3; 4], time reversal focusing imaging [5; 6], diffraction tomography imaging [7], and ultrasonic phased array [8]. Lamb waves are dispersive and multimodal [9]. Moreover, the propagating Lamb waves may include incident, reflected, and converted waves when they encounter a sudden thickness variation [10; 11], such as circular flat bottom holes and rectangular grooves. Obviously, various wave modes make the interpretation of Lamb waves very difficult [12]. In previous literature, the mode identification and separation of Lamb waves have been widely investigated in the time domain, time-frequency domain, and frequency-wavenumber domain [13; 14]. Short-time Fourier transform (STFT) with a sliding window can make the variant distribution of the energy spectrum act as a function of time, which is capable of identifying Lamb wave mode [15]. Frequency-wavenumber analysis based on two-dimensional fast Fourier transform (2D FFT) is a useful approach to distinguish multiple modes. Moreover, the individual Lamb mode can be separated by a 2D band-pass filter [16]. It should be noted that the spatial information is also lost [17]. Compared with the above methods, the local wavenumber approach combined with a spatial window is viewed as an effective way to obtain the space-frequency-wavenumber spectrum. Furthermore, the spatial information can be reserved and the local features of the material can be clearly characterized, including, for instance, the thickness distribution of the material. Many researchers have tried to utilize the above techniques for SHM. [PERSON] et al. adopted a short space 2D Fourier transform to obtain the frequency-wavenumber spectrum at various spatial locations. Consequently, a through-thickness crack of an aluminum plate was discriminated by analyzing the space-frequency-wavenumber spectrum [18], but the size of the crack was not quantified. In another study, [PERSON] et al. proved that spatial wavenumber imaging method was able to characterize the location and length of a crack. However, multiple cracks and cracks with various orientations were not researched in their work [19]. [PERSON] et al. utilized the local wavenumber analysis to characterize the impact damage in composite laminates [20]. The location of a preset delamination was clearly characterized, while its depth was not quantified. [PERSON] et al. reconstructed the thickness distribution of a liquid loaded plate by using an ultrasonic guided wave tomography method based on full waveform inversion (FWI) [21; 22]. It was noted that the remaining wall thickness was reconstructed with high accuracy and resolution. [PERSON] et al. adopted an inverse wave and finite element approach to recover the thickness and density, as well as all independent mechanical characteristics of layered composite structures [23]. However, one drawback of this method is that it is not efficient, as the inversion process is complicated and time-consuming. Therefore, a rapid method for SHM is desired. Air-coupled ultrasonic inspection is a promising non-contact method for the rapid, non-contact inspection of materials [24; 25]. The main advantages of this technique are the absence of any contact and the ability to generate a relatively pure Lamb wave mode when an appropriate incidence angle is adopted. The laser Doppler vibrometer (LDV) based on the Doppler effect has a high spatial resolution in obtaining the propagating Lamb waves [26]. [PERSON] and [PERSON] accomplished the target of actuating the pure A0 mode (anti-symmetric mode) in an isotropic aluminum plate by employing an air-coupled transducer (ACT) and an LDV [27].