# Modeling GPR Wave Propagation in Complex Underground Structures Using Conformal ADI-FDTD Algorithm
## 1 Introduction  
With rapid urban development, the scale of underground pipeline networks has gradually expanded [1, 2]. Pipelines with different functions and purposes are arranged alternately, and the arrangement of the underground pipeline network is very complicated. As detailed distribution information for the entire underground pipeline network cannot be accurately obtained, accidents (e.g., the collapse of old pipelines or the cutting-off of existing pipelines) often occur [3, 4, 5, 6]. In addition, road collapse accidents due to pipeline leaks occur frequently. The quick and accurate determination of the type, size, and location of underground pipelines and hidden diseases around underground pipelines has become necessary in urban road construction and maintenance projects. GPR is widely regarded as one of the most powerful and useful geophysical methods [7]. GPR has the advantage of non-destructive detection and is fast, lightweight, and easy to operate with strong anti-interference and high resolution, etc. compared with other Non-Destructive Testing (NDT) methods such as seismic, sonic, temperature, resistivity, logging, and magnetic exploration methods [8, 9]. Information about the location, size, and dielectric properties of complex underground structures can be obtained by inversion analysis of the GPR echo signals [10, 11, 12, 13]. Through accurate and efficient GPR forward modeling of complex underground structure models, the propagation law of GPRelectromagnetic waves in underground structures can be obtained [14; 15; 16]. This lays the theoretical foundation for inversion analysis and helps to improve the interpretation and processing accuracy of GPR-measured data. At present, the most commonly used GPR forward simulation techniques include the finite element method (FEM) [17; 18], the ray tracing method (RTM) [19; 20], the method of moment (MOM) [21], the finite difference time domain method (FDTD) [22; 23; 24], the pseudospectral time domain method (PSTD) [25], and the symplectic algorithm [26; 27; 28], among others. Although research into GPR forward simulation has achieved fruitful results, these algorithms still have some limitations in terms of computational accuracy and efficiency. For example, the FEM may appear as a \"pseudosolving\" phenomenon during calculation; the RTM cannot consider the dynamic characteristics of the electromagnetic wave of the ground penetrating radar, while the time step of the symplectic algorithm needs to satisfy the Courant-Friedrichs-Lewy (CFL) stability condition, and the computational efficiency is limited. Therefore, it is necessary to propose a GPR forward modeling method with high computational accuracy and fast computational efficiency. The ADI-FDTD algorithm overcomes the CFL stability condition and can employ a larger time step to improve the computing simulation efficiency [29; 30; 31]. The efficiency and accuracy of GPR forward simulation are also related to the modeling method. Some methods are available to improve the accuracy of GPR simulation calculations for circular structures, such as sub-grid technology and conformal grid technology. The sub-grid technique has been widely used to improve the calculation accuracy, but it generates pseudowaves at the interface of the fine and coarse grids [32; 33]. The derivations and calculations for conformal grid technology are simple, making it very suitable for circular underground pipes [34; 35; 36]. In this paper, the efficient and accurate GPR forward models are established to simulate GPR electromagnetic wave propagation in underground structures, employing the ADI-FDTD algorithm and surface conformal technology. We obtained the reflection profile images of multi-pipe models and complex underground pipe models with hidden diseases. By analyzing the profile images, the spatial propagation characteristics of radar waves can be more clearly understood, and the interpretation accuracy of the data can be improved.