## 1 Introduction  
In recent years, the effort in finding \"environmentally friendly\" solutions for the transport of passengers and freight is significantly increased. Urban railway transport is an attractive solution to this aim thanks to its high capacity compared to road traffic. This helps to reduce traffic congestion and air pollution in metropolitan areas. Despite these benefits, the high static loads combined with the high dynamic forces caused by the rail unevenness induce vibrations that may be the cause of several issues. Among others, rail corrugation is one of the major problems related to urban railway lines. It consists of the appearance of a quasi-periodic irregularity on the running surface of the rail that causes ground-borne vibrations, along with secondary noise radiation, and, if it is well developed, degradation of vehicle and track components. The nucleation of rail corrugation is not imputable to a single phenomenon, but different types exist that differ in the characteristics and in the root causes [1; 2; 3]. It is worth mentioning that even if the root causes are different, in any case they are associated with the dynamics of the wheel-rail interaction. In [4], lately reviewed in [5], a classification of the different types of rail corrugation has been proposed. Rail corrugation affects most of the railway lines [4; 5], but it is a critical problem for urban lines, e.g., subways, because the induced vibrations propagate from the track infrastructure to the neighbouring buildings through the soil, causing complaints by the inhabitants. At low frequencies, the discomfort is due to the human sensitivity to whole-body vibration, while at higher frequencies, in the audible frequency range, it is perceived in terms of rumble noise. Mitigation measures addressing the different types of rail corrugation are proposed in [3; 6]. However, the most effective require modifications in vehicle or track architecture and therefore they are less suitable for existing systems. Moreover, a treatment may avoid the occurrence of a specific corrugation class but could favour the formation of a different corrugation type. The usage of different speed profiles, when possible, or friction modifiers typically reduces the corrugation growth rate, but, in general, does not prevent its formation. The most used mitigation action against corrugation is the reprofiling of the rail running surface by means of grinding vehicles when the service is interrupted. This maintenance operation is particularly critical for railway infrastructure managers from both an economical point of view, due to the high initial and operative costs of grinding vehicles, and a managerial point of view, due to the limited time available for such operations. For these reasons, it would be convenient to perform rail grinding only when and where it is strictly necessary. To this aim, the continuous monitoring of rail irregularity along the line will be helpful in locating track sections where corrugation is forming and evolving. Different techniques for the measurement of the longitudinal railhead profile are proposed in [7]. The most promising technique for the measure of the railhead profile at service speeds is the estimation of the irregularity by means of accelerometers mounted in correspondence to the axleboxes of a commercial vehicle. An attempt with a chord-based measurement system is proposed in [8], in which rail roughness is measured at service speed by means of optical sensors mounted on a train bogie. However, the performance of this system depends on the vehicle speed and weather conditions, and its reliability can be significantly affected by dust or other factors in harsh environments. [PERSON] et al. [9] estimate rail roughness by using a hybrid multi-sensor approach in which acceleration signals are combined with acoustic measurements. A model-based estimation of acoustic rail roughness from axlebox acceleration measurements is proposed by [PERSON] and [PERSON] in [10], in which a time domain approach is used.