## 1 Introduction  
The shortage of water resources has constrained the sustainable development of China's economy and the long-term stability of society in recent years. Inter-basin water transfer is one of the effective measures that can be employed to alleviate the regional water shortage in China. The hydraulic tunnel is an important part of the water delivery building. To ensure the safe and efficient operation, a concrete lining is usually imperative. However, factors such as over-excavation, under-excavation, and poor site management during tunnel construction may cause potential defects, such as a lack of thickness, voids at the lining-bedrock interface, and incompact concrete of the tunnel lining. In order to reduce or even eliminate these defects, and make sure that the tunnels are safe after operation, ground penetrating radar (GPR), as an effective detection technique, has been widely applied to tunnel inspection, and has achieved good results [1]. Some voids at the lining-bedrock interface can be found, but the penetration of radar electromagnetic waves is greatly affected by steel reinforcement and water due to the skin effect, which often results in unsatisfactory detection [2; 3]. Moreover, a destructive inspection method like coring will cause unnecessary damage to the lining, which may induce some additional artificial defects to the concrete. Therefore, a new non-destructive inspection method is needed to cover the shortage of GPR. The impact elastic wave is less affected by steel bars and water content because it has the characteristics of a large vibration energy, simple operation, and is convenient for spectrum analysis, etc. Therefore, it is very suitable for the non-destructive inspection of structures. An impact elastic wave is excited by physical means, such as hammer impact or electromagnetic excitation [4]. Since the object being tested is slightly deformed and still in an elastic state, the vibration can be regarded as a result of the elastic wave propagation. The schematic elastic wave propagation generated by impact excitation (Figure 1) mainly includes compression wave P, shear wave S, and Rayleigh wave R [5]. The R-wave is a surface wave propagating along the free surface of the elastic medium. Compared with the P-wave and the S-wave, the R-wave has the largest energy and the smallest attenuation [6; 7; 8]. Therefore, the R-wave (surface wave) signal is easier to collect when carrying out testing on pavement systems [9; 10] and concrete surfaces [11]. Generally, the excitation source is divided into two categories: steady-state forced vibration and transient impact. As a result, the corresponding surface wave analysis is also divided into two types: the steady-state method and transient method. In this paper, only the transient surface wave method is studied since, in the field, it is very hard to implement steady-state excitation on the tunnel lining, especially on the crowned arch. At present, there are two main analytical methods commonly used for transient surface waves: the spectral analysis of the surface wave (SASW) method; and the multichannel transient R-wave methods, like the multichannel analysis of surface waves (MASW) and frequency-wavenumber (F-K) [13; 14]. The SASW method was originally proposed by [PERSON] and [PERSON] in 1985 for a seismic wave test analysis method for detecting the shear wave propagation properties of soil layers and pavements, and later it was adopted in the inspection and safety evaluation of concrete structures [15; 16; 17; 18; 19]. The SASW method includes surface wave generation and processing of transducer received data. The typical test arrangement of the SASW method is shown in Figure 2. An impact is applied to the surface of the medium to be tested by the impact hammer to generate elastic waves with a certain frequency range [20]. The R-waves of different frequencies are superimposed on each other and spread together on the surface of the medium at the same time.