## 2 Theoretical Background  
### Extraction of Multipath Effect  
GNSS monitoring signal is mainly composed of actual vibration, multipath error, and random noise, in which the multipath error is mostly from 0 to 0.2 Hz in frequency domain [10]. The origin of multipath effect is that the received signal in a GNSS receiver may come from several directions and are referred to as multipath signals. In practice, there are objects near the receiver which create replicas of the signal. These replicas go through a longer path than the original signal causing the large propagation delay for these replicas [11]. If the elapsed time is more than a code chip, the receiver usually can cope with multipath disturbance, and multipath signals have a slight effect on the receiver positioning accuracy. Multipath signals with short delay caused by close objects greatly affect the performance of the receiver. Actually, the process of correlation computation between the received signal and the generated reference in the receiver is affected since the received signal is a distorted version of the original signal. Furthermore, phase distortion of the received signal results in pseudorange and carrier phase measurements error and consequently positioning error [12].  
The GNSS observations are always defined as the non-stationary process. Because the GPS multipath error appears in the low-frequency part of the decomposition, the wavelet transform (WT) can be used for suitable resolution acquisition and error extraction. The double difference (DD) residuals are considered to be the input signal by the WT approach. The high frequency components generated by the WT need to denoised. The sampling of orthogonal wavelet on the basis of the discrete wavelet transform in scales is not uniform, and with the increase of the scale, the interval of sampling increases exponentially to the power of 2. The stationary wavelet transform (SWT) is distinct from the standard WT. The filter output by the former is not sub-sampled and the used filters are up-sampled at each level. But the produced result at each level by the latter becomes shorter in length. As a result, SWT can weaken the oscillation effect in DWT. The specific procedures which remove the multipath error of GNSS signal using the SWT method are shown as follows [13].  
The first step of this procedures is applying the SWT to the pseudorange DD residuals using symmelts family as the mother wavelet. The decomposition level is three. In the standard method, the transformation used is the standard WT, whereas in the proposed method the SWT has been used which creates a redundant representation due to its shift invariant property.  
In the second step, the thresholding procedure is used to change coefficients. The threshold can be expressed as [7]  
\[\text{TH}=\frac{\text{MAD}}{0.6745}\sqrt{2(\log L)} \tag{1}\]  
where \(L\) is the length of signal; MAD is absolute median difference of high-frequency coefficients.  
Thresholding is done by determining the method of coefficients reformation and the noise model. The noise model is always defined as the standard Gaussian distribution with zero mean and one variance.  
The third step is reconstructing the coefficients obtained from the previous stage. Since the applied transformation is the SWT, which results in the coefficients redundancy, the reconstruction process has no up samplers.  
The fourth step is applying the extracted multipath to the DD observations. For eliminating the multipath error, the extracted multipath disturbance from the last stage is needed to be applied directly to the DD observations. Then the obtained DD observations eliminating multipath errors are used to the least square method. By this approach, the newly residual shows how efficiently the multipath effect has been removed.