## 1 Introduction  
The rapid growth of the world's freight transportation network has accelerated the development of heavy axle load (HAL) railway transportation. The HAL railway system, which is influenced by dynamic train loads, can adversely impact the reliability of the roadbed, owing to the high amplitude and frequency of the associated loading cycles. Therefore, it is necessary to evaluate whether the existing HAL railway tracks can satisfy the requirements to support trains with a higher speed and larger axle load to improve the transportation capacity of the HAL railway system [1]. In particular, railway tracks can exhibit a deteriorated performance after being subjected to large axle loads or high train speeds, thereby failing to provide the required stability [2]. The service performance of tracks is generally evaluated considering the dynamic response caused by moving trains [3], and the dynamic behavior of the railway system can be examined using numerical simulations. In this regard, the train speed (\(v\)) and axle load (\(F\)) have been identified as the main factors that contribute to the dynamic response of HAL railways [4, 5]. Several numerical models for the railway track have been established to determine the effect of \(v\) and \(F\) on the dynamic responses [5, 6, 7, 8]. However, in these studies, the combined effect of \(v\) and \(F\) was not considered; that is, the results were limited to a single factor (\(v\) or \(F\)). Therefore, it is desirable to develop a united empirical equation of the dynamic response, taking into account the numerical results, which can be used to predict the acceleration and dynamic stress under the combined effect of \(v\) and \(F\). Furthermore, the abovementioned numerical models usually require elastic parameters (i.e., resilient modulus and Poisson's ratio) to be determined for each layer of the substructure. They are both usually obtained by conducting laboratory tests or in situ measurements under those of the loading environment [9; 10]. However, collecting soil samples is often not feasible owing to cost and time limitations and the occurrence of traffic disruption. Moreover, soil samples alone cannot sufficiently indicate the track performance [11; 12; 13]. A method to efficiently and accurately determine the modulus is being considered as the foundation for further development of railway engineering and thus has considerable theoretical and practical significance. One measurement technique, originally developed for road and airfield pavements, involves the use of a light falling weight deflectometer (LFWD), which has been widely applied to investigate the track substructure in the United Kingdom, Ireland, Germany, USA, and Canada [13; 14; 15; 16; 17; 18]. However, a complete description pertaining to the estimation of the resilient modulus of the HAL track substructure in China has not been provided yet. Analyzing the deflection data collected using the LFWD is a prompt and reliable approach to characterizing the properties of the subgrade layers, and extensive efforts have been made to discuss the use of these data as a performance measure for the railway behavior [19; 20; 21; 22]. In addition, several methods to evaluate the performance of railway crossing rails have been presented [23; 24]. However, most of these research efforts were directed toward understanding the effects of the materials, moisture, and seasonal variation. Until now, the problems of the service performance evaluation of HAL railways under the dynamic impact of HAL trains have not been focused on. Therefore, it is necessary to evaluate the stability of HAL railways considering an increase in the v and F of HAL trains. To this end, this paper presents an approach involving LFWD testing to estimate the resilient modulus of the track substructure. In particular, a finite element (FE) model is established using ABAQUS software, in which an inverse analysis subroutine is used to estimate the subgrade modulus from LFWD field test measurements.