## 1 Introduction  
Fatigue cracks, which originate from a damaged precursor at an imperceptible level under repetitive loading, is one of the cardinal reasons for the failure of metallic structures. It is reported that up to 90% of the failures of in-service metallic structures are typically caused by fatigue cracks [1], and the formation of an initial fatigue crack does not necessarily result in immediate failure for the real-world structure [2]. Under cyclic loads, fatigue cracks at the scale of a few micrometers are then accumulated into microcracks by accumulation, which can deteriorate continuously and eventually amalgamate to form macroracks [3; 4]. Under repetitive loads, the macroracks can be further propagated to a crucial level at an amazing rate without sufficient warning, causing detrimental effects on structural integrity and potentially resulting in catastrophic consequences [5]. Therefore, the early perception of small-scale fatigue cracks has become a critical measure to ensure the durability, reliability, and integrity of engineering structures [6; 7]. The majority of current guided wave-based non-destructive testing (NDT) and structural health monitoring (SHM) techniques [3; 8; 9; 10; 11; 12; 13] have been proposed for monitoring crack propagation in different engineering structures, which exploit the variations from temporal features associated with baseline signals. More research studies of temporal signal features primarily focus on the magnitude-based and energy-based signals [14; 15], wave reflections or transmissions [16; 17], energy dissipation [18], and mode conversions [19]. Meanwhile, acoustic emission (AE) technology is widely used for continuously monitoring fatigue cracks, and AE-based fatigue crack evaluation techniques are exploited based on counting the number of signals allured by crack propagation [20; 21]. The scattering and attenuation of ultrasonic guided waves caused by fatigue cracks are applied to quantify the cracks' growth [22; 23; 24]. Although most approaches of temporal features have the capacity to locate fatigue cracks, characterizing the orientation and size of fatigue cracks accurately still remains an extremely challenging task [25], because small-scale fatigue cracks can lead to a feeble signal difference for the damage-scattered waves in the time domain. Therefore, signal features processing and analysis associated with the frequency domain are then extracted to characterize the fatigue crack propagation, because the signal features in the frequency domain are more stable, accurate, and recognizable than those of the temporal domain. The ultrasonic guided waves methods based on damage indices (DIs) have become research hotspots in the field of damage detection over the years, which aim to highlight the variations of structural fatigue cracks. The accuracy and effectiveness of the techniques substantially rely on the means of the defined DI at each path [25], thus, there is an increasing interest in introducing DI for guided wave-based SHM [26]. Based on linear (time of flight (TOF) and energy) and nonlinear temporal features of guided waves, damage indices (DIs) are respectively constructed, which are used to locate fatigue damage near a rivet hole of an aluminum plate [3; 9; 25]. Moreover, the synthetic DI methods are easily interpretable, which can intuitively reflect the overall health status in the inspected structure. A hybrid DI characterization scheme for ultrasonic guided waves is developed and experimentally validated in aluminum plate specimens, which can be used to evaluate the statistical distribution of fatigue cracks under uncertainties and update the prognosis results. However, according to the aforementioned literature survey, damage indices are used to locate cracks and identify the existence of multi-scale cracks in practice [25]. The crack growth is a gradually pathulous process, the ultrasonic guided waves DI approach has been rarely employed to characterize the crack propagation quantitatively.