## 1 Introduction  
Rail corrosion is one of the common faults occurring in plain line railway tracks and has attracted notable attention in recent years [1]. [PERSON] [2] evaluated the corrosion risk, according to chemical simulation of corrosion formation and propagation at the rail base. Visual inspection techniques are commonly used in corrosion inspection processes [3; 4; 5; 6]. However, most inspection focuses on the accessible surfaces of the rail. There are limitations in the current inspection techniques, especially in regard to corrosion of the underside of the rail. For example, it is difficult to detect corrosion through 3D visual inspection where a mapping laser and optical scanner cannot be mapped, due to the location of the fault as it is below the rail and not accessible. Additionally, electromagnetic methods can be limited in extracting information from the underside of the rail where their signals are unable to pass through the whole rail body, due to the skin effect [7] and other attenuation. In this paper, acoustic inspection is a non-invasive technique that can be used widely and provides accurate results. Acoustic techniques have been used in modern railway inspection, but only in limited applications, such as bearing and gearbox monitoring, in which there has been rapid development in recent years due to the capabilities and effectiveness of acoustic techniques in detecting bearing and gearbox faults in the early stages [8; 9; 10]. The sound generated from wheelsets passing through rail discontinuities, such as crossing noses and insulted rail joints (IRJs), can be related to their structural performances, which can potentially be used in fault detection. However, it is challenging to obtain data from vehicles passing damaged rail discontinuities for safety reasons, and, therefore, faulty track pieces have been in laboratory conditions for this kind of inspection. The signal generated by hammer impact is used for the analysis (impact response signal). This approach has previously been used for classifying faults on railway tracks [11]. Signal decomposition techniques have commonly been used in signal processing methods [8; 12; 13; 14; 15]. Variational Mode Decomposition (VMD) is a self-adaptive signal de-composition method, proposed in 2014 [16], which has been adopted in inspection applications, particularly for rotational machines. [PERSON] [17] proposed an improved VMD method, by combining VMD with a deep convolutional neural network (DCNN), for the detection of rolling bearing element faults, which enhanced the visibility of the fault feature and, therefore, improved the accuracy of diagnosis. [PERSON] [18] considered the feasibility of extracting fault features from turbine bearings through a combination of VMD and particle swarm optimisation (PSO). This application has the advantage of being able to decompose complicated signals, compared to conventional Empirical Mode Decomposition (EMD). The authors in [19] proposed an application combining VMD and permutation entropy to identify different fault types within bearings with high classification accuracy. [PERSON] [20] considered the application of inspection encoder signals from gearboxes. The signals were decomposed using an advanced VMD method combined with a sparse representation to extract fault information. [PERSON]'s proposed method both suppressed noise and enhanced fault impulses. Considering these works, VMD is often applied to signals emanating from rotational machines. However, the features from impact response signals are somewhat different. Considering the feasibility of impact response signals decomposed using VMD, [PERSON] [10] demonstrated that VMD is able to decompose impact response signals from railway tracks with high classification accuracy. However, the faults considered did not include corrosion. This paper, therefore, used VMD for the application of acoustic inspection to the detection and classification of corrosion on the underside of rails.