# Rail Diagnostics Based on Ultrasonic Guided Waves: An Overview
## 7 Land-Based Systems Implementation-Evolution: Early Rail Defect Detection Capability
### Adaptive SAFE Model for Rail Parameter Estimation
#### 7.5.2 Possible Solutions  
Material and geometric parameters are implicit in the computation of the dispersion curves. For this reason, inferring these parameters from such curves requires solving an inverse problem through an iterative approach. Solving this inverse problem becomes easier if the corresponding forward problem, consisting of the computation of dispersion curves, can be solved efficiently and if the number of rail parameters to be estimated can be reduced.  
The use of the _semi-analytical finite element_ (SAFE) method for efficiently solving the forward problem is investigated in Refs. [52; 57]. In these manuscripts, a set of parameters describing the geometry and material properties of a worn rail has been determined. This set is employed to develop a SAFE model for computing the propagation characteristics of worn rails. Wear and grinding of the railhead can be represented through the three geometric parameters shown in Figure 19. An adaptive mesh model has also been developed by means of parametric equations to represent material removal. This allows an efficient automatic modification of the geometry and the mesh of the considered rail.  
The use of the SAFE method requires the knowledge of different properties of the employed material, such as its _elastic modulus_\(\mathbf{E}\) and its _density_\(\mathbf{\rho}\), or its _Poisson's ratio_\(\mathbf{\
u}\). The evaluation of the dispersion curves of a waveguide through this method requires  
Figure 19: Representation of the parameters useful to identify the presence of a worn rail.  
solving an eigenvalue problem. Material properties are usually unknown to an acceptable accuracy, and their value is dependent on environmental conditions. In the case of an isotopic elastic medium, it is shown that \(\mathbf{E}\) and \(\mathbf{\rho}\) can be combined in a single parameter, namely the _longitudinal speed of sound in the medium_, \(\mathbf{c}=\sqrt{E/\mathbf{\rho}}\). Therefore, only two parameters, \(\mathbf{c}\) and \(\mathbf{v}\), are required instead of three (i.e., \(E\), \(\mathbf{\rho}\), and \(\mathbf{v}\)): this allows for saving computation time. Computational efficiency can be further improved by solving the eigenvalue problem in terms of the parameter \(\mathbf{\beta}=\omega\mathbf{c}\), i.e., of the ratio between the frequency \(\mathbf{\omega}\) and the longitudinal speed of sound \(\mathbf{c}\). Thus, only a single set of dispersion curves needs to be computed for each value of the couple \((\mathbf{c},\mathbf{v})\): if \(\mathbf{c}\) changes, but \(\mathbf{v}\) remains constant; the dispersion curves need only to be scaled according to the variation of the parameter \(\mathbf{\beta}\).  
Experimental results have proven that various propagating modes could be detected on field. This has allowed for achieving a proper tracking of mode shapes computed through the SAFE method by using different material and rail geometric properties. The _modal assurance criterion_ (MAC) has been adopted to accomplish the tracking task. More specifically, the mode shapes produced by the SAFE method in the presence of different input parameters have been correlated with three reference groups of the mode shapes identified manually on the basis of field measurements, as illustrated in Figure 20 [57].  
Finally, it is worth mentioning that the technique developed in Ref. [57] allows for identifying which set of simulated dispersion curves best fits the experimental spectrograms computed in the absence of prior knowledge about the distance of the reflectors from the transducer. Experimentation results on the field have proven that this technique performs well.