**1. Introduction**

It is widely accepted that a track modulus, and its variations, are indicators of subgrade conditions [1–5]. A track modulus is a measure of the vertical stiffness of the rail foundation and is defined as the ratio of the vertical supporting force per unit length of rail to the vertical deflection [1]. A practical way to assess the track modulus is to measure the rail deflection under specified loads [6–8]. Measured deflections can be correlated to the track modulus using mathematical equations. Two methods are available to measure rail deflections: trackside measurement techniques and on-train approaches. Trackside measurement techniques are used to measure the rail deflection at specific locations under specified static loads or moving loads [9]. Although these techniques provide accurate estimations of track stiffness, they are laborious and time-consuming, especially when multi-point measurements are required. On the other hand, on-train measurement systems allow the measurement of rail deflections over long distances and thus provide a good overall evaluation of the entire railway network [10–15]. Comprehensive analysis is typically needed to investigate the relationship between deflection measurements from on-train systems and track modulus [16,17].

The real-time vertical track deflection measurement system (known as MRail System) developed at the University of Nebraska–Lincoln, under the sponsorship of the Federal Railroad Administration (FRA), has become more popular in recent decades [10–12]. The system computes relative vertical deflection (*Yrel*) between the rail/wheel contact plane, and the rail surface at a distance of 1.22 m from the nearest wheel to the sensor system. The MRail system has been tested over different railway lines in the USA and Canada for evaluating track conditions [18–21]. Results from the MRail field tests show that the system not only has the potential to identify the local track problems, i.e., muddy ballast, degraded joints, crushed rail head, broken ties, but also provides an opportunity to map the subgrade condition and assess the track performance along the railway line [22–25].

In addition to the experimental studies, different numerical models have been used to investigate the relationship between track modulus and *Yrel* data, and numerical approaches have been proposed to estimate the track modulus from *Yrel* [21,26]. The current study aims to propose a new and advanced approach for estimating track modulus statistical properties from *Yrel* data more accurately compared to previous studies. First, the details of the MRail system are briefly presented, and the numerical models developed by others and their shortcomings are discussed. Then, artificial neural networks (ANNs) are explained as the main tool to investigate the relationship between track modulus and *Yrel* data in this paper. Different methods for training the ANNs are used, and the effectiveness of the trained ANNs are investigated using error measurement parameters such as the coefficient of determination (*R*2), the root mean square error (RMSE), and mean absolute percentage error (MAPE). Suitable signatures of *Yrel* data are identified by conducting both statistical and frequency analysis. Feedforward neural networks are proposed as a function approximation technique to estimate the track modulus average (*UAve*) and standard deviation (*USD*) from *Yrel* data. To further investigate the effectiveness of the ANNs for estimating the track modulus, noisy finite element models (FEM) datasets are employed for training the ANNs. The accuracy of the track modulus estimations using these ANNs is also investigated using *R*2, RMSE and MAPE.

#### **2. The Sti**ff**ness Measurement System and Numerical Simulations**

#### *2.1. MRail Measurement System*

The MRail system was originally developed at the University of Nebraska–Lincoln under the sponsorship of the Federal Railroad Administration (FRA) [10–12]. The system measures the relative vertical deflection (*Yrel*) between the rail surface and the rail/wheel contact plane, at a distance of 1.22 m from the nearest wheel to the acquisition system (Figure 1a). The sensors consist of two laser lines and a digital camera mounted on the side frame of the rail car (Figure 1b). The laser system projects two curves on the rail surface, whose minimum distance (*d*) is captured by the camera (Figure 1c). Subsequently, the distance between the camera and the rail surface (*h*) is computed by converting *d*. Finally, the relative deflection *Yrel* is calculated by subtracting *h* from (*Yrel* + *h*), the fixed distance between the rail/wheel contact plane and the camera.

The MRail system can measure the deflection at different sampling rates with the speed up to 96 km/h (60 mph). The Winkler model and the finite element models have been used to estimate the track modulus from *Yrel* [24].

**Figure 1.** Demonstration of the MRail system (real-time vertical track deflection measurement system) for *Yrel* measurements: (**a**) the measurement system on a rigid frame; (**b**) the sensor system; and (**c**) the projections of the laser lines on the railhead.
