# Snow Depth Variations in Svalbard Derived from GNSS Interferometric Reflectometry
## 2 Data and Methods
### Method
#### 2.2.2 Improved Approach Based on Wavelet Analysis  
Commonly used methods for improving the signal-to-noise ratio include the empirical modal solution, independent component analysis, wavelet decomposition and so on. However, under low signal-to-noise ratio conditions, there may be aliasing effects with the empirical modal solution, the autocorrelation matrix of the signals may have zero eigenvalues, and the whitening process cannot be completed for independent component analysis. By contrast, the multi-resolution characteristics of wavelet decomposition can accurately represent the local information of the signal in the time and frequency domains and overcome the shortcomings of the short-time Fourier transform at the single-resolution scale. Wavelet decomposition is now widely used in time-frequency analysis, signal-to-noise separation and weak-signal extraction [28; 29]. In this study, before performing LSP spectral analysis on the SNR sequence, wavelet decomposition was carried out to reduce the influence of the noise power. [PERSON] and [PERSON] proposed the use of wavelet decomposition to express the signals that meet the conditions as a series of successive approximation expressions for the first time, and each of these expressions is the smoothed form of the original signal, corresponding to different resolutions [30]. Wavelet analysis has the advantage of multiple resolutions, and both the time and frequency windows can be dynamically adjusted according to the specific form of the signal. Multi-resolution analysis is a set of functions that construct closure approximation L\({}^{2}\)(R), and the set of functions constitutes a normalized orthogonal basis. For any given signal, its expansion in wavelet space can be described as  
\[f(t)=\sum_{j\in\mathbb{Z}}f_{wj}=\sum_{j\in\mathbb{Z}}\sum_{k\in\mathbb{Z}} \Big{\langle}f(t),\psi_{j,k}(t)\Big{\rangle}\psi_{j,k}(t) \tag{1}\]  
where \(Wj\) is the orthogonal complement of the closed subspace Vj on L\({}^{2}\)(R); \(\mathbb{V}(t)\) is a function in L\({}^{2}\)(R). The corresponding multi-resolution analysis is  
\[f(t)=\sum_{j\in\mathbb{Z}}\Big{\langle}f(t),\varphi_{j,k}(t)\Big{\rangle} \varphi_{j,k}(t)+\sum_{j\in\mathbb{Z}^{\prime}\leq j}\sum_{k\in\mathbb{Z}} \Big{\langle}f(t),\psi_{j^{\prime},k}(t)\Big{\rangle}\psi_{j^{\prime},k}(t) \tag{2}\]  
Figure 2: LSP spectral analysis results for the SNR sequences for GPS PRN 19 on February 28, 2018. A peak corresponds to a main frequency, but two similar peaks mean the failure of LSP in obtaining the main frequency. where \(\varphi\) is the scale function, \(j\) and \(j^{\prime}\) are the resolution scale, \(j\) is the maximum resolution scale, and \(k\) is the translation scale. For the signal-to-noise ratio sequences after trend removal, the Mallat algorithm was used for multi-resolution analysis, and the features of interest are extracted by reconstructing the original signal structure [31]. The corresponding frequencies of different resolution scales can be calculated as  
\[w_{j}=\frac{1}{\|\varphi_{j,k}\|_{0}^{2}}\int w\big{|}\varphi_{j,k}\big{|}^{2}dw \tag{3}\]  
\[\Delta w_{j}=\frac{1}{\|\varphi_{j,k}\|_{0}^{2}}\bigg{\{}\int(w-w_{j})^{2} \big{|}\varphi_{j,k}\big{|}^{2}dw\bigg{\}}^{\frac{1}{2}} \tag{4}\]  
where \(w\) is the center frequency and \(\Delta w\) is the frequency radius. Since \(w\)j is an orthogonal complement of Vj, \(w\) is also an orthogonal sequence set. After a comparative experiment, a compactly supported orthogonal wavelet decomposition was performed with a wavelet order of 6 [32]. The results of the discrete wavelet decomposition of the SNR sequences of the PRN17 satellite on February 28 of 2018 are shown in Figure 3.  
Different resolution scales correspond to completely different frequencies, and the higher the resolution scale, the narrower the frequency window. The coefficients obtained by wavelet decomposition correspond to different signal frequencies, such as the receiver thermal-noise frequency, environmental-noise frequency and signal frequency related to snow depth.