## 1 Introduction  
On a traditional railway track, a fishplate and bolt, as shown in Figure 1, are used to join rails [1]. The large dynamic force generated by geometric discontinuities between a wheel and the rail at joint gaps is one of the weak points in track structure. However, as shown in Figure 1, rails of standard length can be welded into rails of extended length, which is one important measure towards eliminating rail joints. Flash welding and thermite welding are mainly used in rail welding. However, absolute smoothness of the geometric surface of the rail weld cannot be attained due to limitations in welding technology and improper operation, etc. If the temperature of the rail surface and adjacent area is lower than the melting temperature during flash welding, achieving complete contact between the melt and parent metal is hard and crystalline lenses, micro-pores, and non-metallic impurities will appear in the welds during forging [2]. In addition, the strength, hardness, and micro-structure of the welding materials are different from that of the parent metal [3], and unaligned ends of welded rails, inequality of grinding, and welding parts by hand will cause irregularities on the surface of the rail [4]. [PERSON] and [PERSON] [5] carried out a detailed survey on geometric irregularities at rail welds on high-speed railways in China. They found that a short-wave irregularity with a wavelength range from 0.05 to 0.39 m was the main form. Initiation and development of such an irregularity can be from high-frequency excitation, which finally loses its energy by dissipation in the form of plastic deformation and other processes related to the growth of track damage, as reported by [PERSON] [6], [PERSON] [7], and [PERSON] and [PERSON] [8]. Therefore, controlling the size of the short-wave irregularity of welds is of great importance to ensure secure, stable, and economic operation of high-speed railways. [PERSON] [9] investigated the influence of weld defects with different wavelengths on the wheel/rail force and proposed safety limits under different speeds based on the dynamics index or dynamic factors. [PERSON] and [PERSON] [10,11] evaluated the quality of rail welds using a geometric gradient, which can be used for considering any geometric form of the weld. Related standards have also been determined on the basis of numerical calculations and engineering experience. The precondition of carrying out the evaluation of wheel-rail force at rail welds stated in [9, 10, 11] is to obtain its geometric size, which is definitely time consuming and labor intensive. For effective measurement of irregularities in the tracks, detecting vertical axle box acceleration (hereinafter referred to as ABA) is a simple and economical method. An accelerometer sensor can be easily installed on the axle box (see Figure 2) without limitations of the vehicle's running speed [12,13]. To reflect on the health condition of track systems, many scholars have, therefore, focused their studies on ABA. [PERSON] [14] developed a set of indoor rolling test devices that was used for simulating the response of ABA under low-speed conditions, which ranged from 3.5 to 15 km/h, when there were defects on the wheel/rail surface. Obvious fluctuations in ABA were generated at the defects that were confirmed via time-frequency technology according to research findings to be damage on the wheels or rails. [PERSON] and [PERSON] [15] carried out a series of ABA tests targeted at monitoring the health of some preliminary damage (e.g., rail squats, rail pad degradation, and fastening cracks) occurring at weld joints on site; the results showed an obvious difference between rail welds that were in good and poor condition. On this basis, [PERSON] [16] put forward three approaches to improve the success rate of monitoring. However, the methods are currently applied on lines of ordinary speed (that is, 140 km/h). [PERSON]'s group [17] conducted abundant tests on high-speed railway lines in China.