## 2 Theoretical Background
### Denoising Random Noise  
The relationship between the monitored data and real deformation signal can be expressed as:  
\[x(t)=s(t)+n(t) \tag{2}\]  
where \(x(t)\) is recorded data, \(s(t)\) and \(n(t)\) are the true signal and noise, respectively. Because noise is ubiquitous and represents a highly undesirable and dreaded part of any data, many data analysis methods were designed specifically to remove the noise and extract the true signals in data, although often not successful.  
The _Empirical Mode Decomposition_ (EMD) has been proposed recently as an adaptive time-frequency data analysis method. The EMD can decompose any data into a set of _Intrinsic Mode Function_ (IMF) components, which become the basis representing the data. Such an EMD method is not limited by the Heisenberg uncertainty principle, and can get relatively high frequency resolution. Also because of the adaptive nature of the basis, there is no need for harmonics; therefore, EMD is ideally suited for analyzing data from nonstationary and nonlinear processes. The basic process of decomposition is shown in the Refs. [14; 15].  
Through EMD, the signal containing noise \(x(t)\) is decomposed into the intrinsic mode IMF\({}_{i}\) and residual component \(r_{n}\) from high frequency to low frequency, that is  
\[x(t)=\sum_{i=1}^{n}\text{IMF}_{i}(t)+r_{n}(t) \tag{3}\]  
where \(r_{n}(t)\) is the residue of data \(x(t)\), and IMFs are simple oscillatory functions that vary with amplitude and frequency.  
The filtering method relies on the basic idea that most of the important structures of the signal are often concentrated on the lower frequency ones (last IMFs) and decrease toward high-frequency modes (first IMFs). It is generally believed that high-frequency components are corrupted by the noise, while low-frequency components are subjected to little impacts of the noise. According to this idea, there is a mode, after which, monitored data are dominated by the real dynamic deformation signal, whereas the previous modes are high-frequency components dominated by the noise. It has been shown that based on partial reconstruction of relevant modes, EMD performs signals filtering in an adaptive way [16]. However, it still raises two questions on how to select such modes in an efficient way, and some IMFs which contain noise and signal both are abandoned directly. The authors in [17; 18] use a correlation-based threshold to discriminate between relevant and irrelevant IMFs. Generally speaking, the model of Gaussian white noise is usually employed to simplify the random noise in GNSS signal. The autocorrelation function of the white noise at the zero-point equals to one, and although it is not equivalent to zero at other points, the values of autocorrelation functions of such points are lower compared with that of the zero point. In the paper, based on the characteristics of Gaussian white noise model, autocorrelation function is employed to judge the number of IMF truncation, and then all components that have been processed and unprocessed are reconstructed to gain the denoised GNSS signal, \(x^{\prime}(n)\), which is expressed as  
\[x^{\prime}(n)=\sum_{i=1}^{k}\text{IMF}_{i}+\sum_{j=k+1}^{N}\text{IMF}_{j} \tag{4}\]  
where \(k\) is the order of the dominant mode of noise determined by autocorrelation function. IMF\({}_{j}\) whose order is lower than \(k\) should be filtered by wavelet soft threshold.