## 1 Introduction  
In recent years, with the vigorous development of nondestructive testing techniques for pavements, falling weight deflectometer (FWD) has been widely applied. Simultaneously, the inversion analysis of the dynamic modulus of pavement structures has also become one of the most critical techniques used in their service performance evaluation [1]. The method for solving the dynamic responses of a multi-layered pavement under impact loads is limited by the boundary conditions and computational efficiency; thus, in the early development stage of FWD instruments, most inversion software was based on the quasi-static analysis method [2]. However, this method cannot simulate the dynamic effect of vehicle loads. Moreover, the static inversion method is prone to causing the phenomenon of \"modulus transferring\", which leads to the failure of the inversion results [3]. Although the dynamic effect of loads can be considered in the dynamic response analysis of the pavement structure system by the finite element method (FEM), its solution process requires meshing the pavement structures, which greatly reduces its solution efficiency and seriously affects its application in the inversion of the dynamic modulus of the pavement structures [4, 5]. In recent years, with the deepening of research on the dynamic responses of layered semi-infinite media, increasingly more dynamic response algorithms have emerged, most notably the integral equation transformation method [6; 7; 8; 9]. These algorithms can obtain the dynamic displacement and stress responses of layered soil under dynamic load in the frequency domain and can reveal the dynamic characteristics of the layered soil. However, when solving the displacement responses of road surfaces under FWD loads, it is necessary to solve the infinite integral problem; thus, the solution efficiency is greatly reduced. As an algorithm for solving the dynamic responses of layered soil under axisymmetric loads, the spectral element method (SEM) not only can consider the dynamic effect of FWD loads to obtain accurate calculation results, but also has high solution efficiency [10; 11]. Thus, it has been successfully applied to the back-calculation of the dynamic modulus of pavement structures [3; 12; 13; 14; 15; 16; 17]. With the development of forward algorithms for the dynamic responses of layered pavement structures, a variety of dynamic modulus inversion software has been developed based on nondestructive testing techniques using FWD equipment. However, most existing inversion software is based on the peak deflection of the road surface, which only reflects the overall bearing capacity of the pavement structure. The deflection basin displacement time-history curve obtained by FWD testing equipment contains comprehensive pavement structure information, which cannot be completely reflected by its peak deflection. In the inversion process, the insufficient input conditions usually lead to the non-unique problem of parameter inversion. It is thus difficult for the inversion results to objectively characterize the actual performance of each structural layer of the pavement [13; 14; 15]. [PERSON] et al. [13] used SEM as the forward model, and used the backpropagation neural network (BPNN) and real-coded multi-population genetic algorithm to predict the structural parameters of pavements. While the input conditions of the pavement mechanical model were increased (the interlayer contact conditions and transverse isotropic properties of materials), the non-unique problem could not be solved by relying only on the maximum deflection of the road surface. [PERSON] et al. [3] also used SEM as a forward analysis method, constructed an optimization objective function using the measured deflection time-history curve and theoretical calculation results, and carried out the dynamic inversion of the mechanical parameters of a pavement via numerical optimization methods.