# Spatiotemporal Monitoring and Evaluation Method for Sand-Filling of Immersed Tube Tunnel Foundation
## 2 Elastic Wave Monitoring for Sand-Filling of Immersed Tube Tunnel  
### Elastic Wave Monitoring Method  
Elastic waves propagate in three directions in three-dimensional media. According to wave propagation theory, in a layered medium of elastic semi-infinite space, when the surface of the medium is subjected to instantaneous impact, the point of action produces instantaneous vibration. The vibration generated by the point of action spreads far away and forms an elastic wave. When the internal composition of the medium changes, the elastic wave is reflected and transmitted where the wave impedance is different, resulting in a change in the composition of the elastic wave. By analyzing the collected data of elastic waves, the information of the target can be obtained according to the time domain, frequency domain, or time-frequency domain characteristics. The elastic wave monitoring method involves using the elastic wave excited by the impact hammer hitting the surface of the medium, analyzing the vibration of the measuring point to obtain the lower medium information, and comparing the change in the medium information for engineering monitoring. A sketch of the elastic wave monitoring method for sand-filling in an immersed tunnel is shown in Figure 1. The sand-filling model of the immersed tube tunnel can be divided into three layers: the concrete, sand-filling, and gravel cushion layers. The sand-filling engineering of immersed tube tunnels is the dynamic process of filling the sand-filling layer to form a sand deposit. The diffusion radius of the sand deposit increases with the sand-filling progress. The monitoring of the sand-filling state is realized by monitoring the diffusion radius \(\text{R}_{i}\)'. To monitor the diffusion radius of the sand deposit, a series of impact-and-receive operations are performed on the free surface of the model. When \(\text{I}_{\text{i}}\) and \(\text{R}_{\text{i}}\) represent the locations of a source-and-receiver set for the elastic wave measurement, after the elastic wave is excited by \(\text{I}_{\text{i}}\), it is received at the receiving point \(\text{R}_{\text{i}}\), where the distance to \(\text{I}_{\text{i}}\) is 10 cm. The elastic wave reflects the formation information of where \(\text{R}_{\text{i}}\) is located. For immersed tube tunnel sand-filling, the thicknesses of the lower gravel layer and the upper concrete layer do not change. The elastic wave during the sand-filling process reflects the development information of the middle sand-filling layer, which is called the tunnel foundation cushion filling state. The received elastic waves are principally analyzed in the following three aspects to obtain the feature parameters. Time domain analysis  
The amplitude of the received wave reflects its energy. The change in the amplitude of the waveform reveals the variation of wave impedance inside the model, according to which the information of the medium inside can be further judged. To characterize the amplitude of the received wave more intuitively and systematically, the integral of amplitude over time is used in the time domain. To avoid the impact of the hammering force on the wave energy during excitation, the square of the maximum elastic wave amplitude is regarded as the information that can reflect the energy of the hammering force, which is divided in the characteristic parameters. The energy parameters \(PP_{i}\) are defined as the feature parameters of the elastic wave in the time domain as follows. \[PP_{i}=\frac{\int_{t1}^{t2}|f(t)|dt}{\max\limits_{0<t<t3}f(t)^{2}} \tag{1}\]  
where \(t\) is time, \(t_{1}\) is the starting time, \(t_{2}\) is the ending time, \(t_{3}\) is the lasting time, and \(f(t)\) is the function of the time-history velocity wave. Frequency domain analysis  
For multilayered media, a certain predominant frequency of the received wave and its spectrum reflects the characteristics of elastic waves in the frequency domain. A fast Fourier transform is conducted on the received elastic wave to obtain the spectrum curve.