# The Fractal Characteristics of Ground Subsidence Caused by Subway Excavation
## 1 Introduction
[21] employed the mercury intrusion pore method and low-field nuclear magnetic resonance method to quantitatively analyze the pore fracture structure and its fractal features. [PERSON] et al. [22] employed the box-counting method and Sierpinski carpet analytical solution to analyze the fractal features of shale pore structure at multiple scales. Although previous studies have made progress in understanding the relationship between fractal dimensions and geological deformation processes, the accuracy and reliability of the analysis are often affected by noise, missing data, and outliers in monitoring data. Fractal interpolation, as an interpolation method based on fractal theory, addresses these challenges by filling in missing data points while correcting and optimizing abnormal data. This method effectively deals with irregularities and missing data. [PERSON] et al. [23] used R/S and V/S analysis to derive the Hurst index of tunnel surface deformation time series based on fractal theory. They then applied fractal interpolation to predict the trend of tunnel deformation evolution. [PERSON] et al. [24] proposed a fractal interpolation method to fill in missing data using a sequence of self-similar features. [PERSON] [25] introduced the Cuckoo Search-Designated Fractal Interpolation Functions (CS-DFIFs) method and the CS-DFIFs-winners combining (CS-DFIFs-WC) method to estimate missing values. [PERSON] et al. [26] proposed a method for accurately recovering seismic data using the local fractal recovery method, which enables easy and accurate recovery of the intrinsic structure associated with the seismic data. In addition, the time series data for surface settlement are often affected by various types of noise, posing challenges for accurately extracting fractal features. Wavelet noise reduction emerges as an effective signal processing method capable of removing random noise from the data while preserving the main features of the signal, making it a more effective approach [27; 28]. [PERSON] et al. [29] analyzed the dynamic deformation data of high-rise buildings monitored in the field using wavelet noise reduction, concluding that wavelet analysis has an advantage over traditional signal processing techniques in non-stationary monitoring signal processing. [PERSON] et al. (2019) observed a significant enhancement in the first-to-first characteristics of signals after noise reduction and compared the spectra of microseismic signals before and after wavelet noise reduction. [PERSON] et al. (2019) employed wavelet noise reduction to preprocess hydropower station landslide monitoring data, revealing that analyzing the deformation time series after noise reduction better describes the deformation trend of landslides. Regarding foundation tunnels, [PERSON] et al. (2019) utilized wavelet noise reduction on data related to underground station settlement and combined it with a BP neural network to predict underground settlement, resulting in improved accuracy. [PERSON] et al. (2019) employed a combination of wavelet analysis and neural networks to predict the settlement of deep foundation pits, achieving an error between the predicted and actual values of from 1% to 5%, demonstrating the effectiveness of wavelet noise reduction in signal processing. [PERSON] et al. (2019) applied wavelet noise reduction to a section of the Wuhan Yangtze River Tunnel online monitoring data, effectively reducing noise while retaining the original data mutation and aligning the monitoring data more closely with the real trend of change. [PERSON] (2019) also utilized wavelet transform to reduce noise in shield tunnel monitoring data, providing reliable results for diagnosing the tunnel's health. The existing research faces two main issues. Firstly, surface settlement data are often affected by noise interference, missing data, and errors resulting from monitoring instruments, manual input, and environmental factors. Secondly, there is a lack of in-depth studies that utilize surface settlement deformation information to guide monitoring work.