# Initial Results from SQUID Sensor: Analysis and Modeling for the ELF/VLF Atmospheric Noise
## 3 Data Preprocessing
### Baseline Correction  
Baseline wandering was often observed in low-frequency measurement ([PERSON] et al., 2016; [PERSON] et al., 2016), which would obscure and even deteriorates the analysis result. Consequently, it is necessary to correct the baseline of the observations to extract buried signals ([PERSON] et al., 2016). As the atmospheric noise often exhibits non-stationary characteristics, wavelet based methods can provide multi-resolution analysis for the measured magnetic field signal. Recently, [PERSON] et al. (2016) proposed two noise reduction techniques to remove atmospheric noise from airborne transient electromagnetic data via a stationary wavelet transform. [PERSON] et al. (2016) adopted a combined wavelet transform method to reduce the white Gaussian noise and baseline drift jointly for the airborne transient electromagnetic data. Furthermore, [PERSON] et al. (2016) suppressed baseline drift effectively by using a wavelet-based correction algorithm. Consequently, wavelet transform is adopted to correct baseline drift by choosing appropriate wavelet basis and decomposition level in this paper. The two-scale equations are given by  
\[\phi(t)=\sqrt{2}\sum_{n}h_{0}(n)\phi(2t-n), \tag{1}\]  
Figure 2: Magnetic noise spectra of the mobile SQUID system measured inside a magnetically shielded room (MSR). \[\psi(t)=\sqrt{2}\sum_{n}h_{1}(n)\phi(2t-n), \tag{2}\]  
where, \(\phi(t)\) is the scaling function, \(\psi(t)\) is the wavelet function. The wavelet transform can be regarded as a filter bank and \(h_{0}\) and \(h_{1}\) are filter coefficients. The baseline wandering can be corrected by discarding the estimated low frequency components decomposed by the wavelet transform. As the length of real world signal is finite, the wavelet denoising method would cause distortion on the boundary of the signal. However, it was ignored in many wavelet-based filtering methods [20; 21]. In order to alleviate the adverse boundary effect by traditional wavelet method, we extend the data at both ends of the according time series with a mirror extension. The corresponding waveform processed by wavelet methods, adopting \"sym8\" base [21] and 14 levels are shown in Figure 3. As the sampling rate is 100 kHz, the frequency components lower than 3 Hz will be filtered. It is shown that wavelet methods can provide perfect baseline correction results for our data. Moreover, the boundary effect caused by traditional wavelet methods can be eliminated by extending  
hence only the first \(K\) harmonics are considered in this paper. The estimate of the harmonics \(p(t)\) from \(y(t)\) can be derived by a least square (LS) estimation  
\[\text{minimize}\quad(Y-XP)(Y-XP)^{T}. \tag{5}\]  
The fundamental frequency \(f_{0}\) is obtained firstly by setting \(K=1\), followed with the estimation of \(P\)  
\[\hat{P}=(X^{T}X)^{-1}X^{T}Y. \tag{6}\]  
When \(p(t)\) is obtained by (4) and (6), \(s(t)\) can be derived by subtracting \(p(t)\) from \(y(t)\)[24]. It can be found that \(p(t)\) largely depends on the estimate of \(f_{0}\). However, \(f_{0}\) is time-varying due to the drifting nature caused by generator and power plant design [3]. Consequently, the data should be divided into frames before LS estimation. A smaller length of the frame would lead to a better stationary characteristic of the data. While the performance of the LS estimation deteriorates as the number of samples decreases. Considering all these factors, the length of the frame \(\Delta t=0.2\) s and the number of harmonics \(K=20\) were chosen. The waveform and the corresponding spectra of the processed data are shown in Figure 5. It can be seen in Figure 5b that the background noise is quite low in comparison to measurements in urban area and there are only odd harmonics in the measured data. As a result, the distortion of the spectra can be diminished by designing \(X(t)\). The gray line denotes the spectra of the observed atmospheric noise before baseline correction.