Multiscale Analysis of GPS Time Series from Non-decimated Wavelet to Investigate the Effects of Ionospheric Scintillation

Abstract

ABSTRACT. Due to the numerous application possibilities, the theory of wavelets has been applied in several areas of research. The Discrete Wavelet Transform is the most known version. However, the downsampling required for its calculation makes it sensitive to the origin, what is not ideal for some applications, mainly in time series. On the other hand, the Non-Decimated Discrete Wavelet Transform is shift invariant, because it considers all the elements of the sample, by eliminating the downsampling and, consequently, represents a time series with the same number of coefficients at each scale. In the present paper, the objective is to present the theorical aspects of the a multiscale/multiresolution analysis of non-stationary time series from non-decimated wavelets in terms of its implementation using the same pyramidal algorithm of the decimated wavelet transform. An application with real time series of the effect of the ionospheric scintillation on GPS satellite signals is investigated. Results obtained in this investigation of the time series of the S4 scintillation index indicate the presence of a pattern that is repeated in the series on consecutive days. Through decomposition in multiscale by wavelets, this periodic daily effect could be identified in smoother scales of the wavelet periodogram, estimated and corrected from the original time series. Therefore, this paper presents an innovative methodology for the analysis of the effects of ionospheric scintillation on GPS signals, and take a first step toward separating the effect of scintillation in relations the other effects that may influence its analysis.

Non-decimated Wavelet Transform for a Shift-invariant Analysis

“…To illustrate the shift-invariance of the NDWT also in the real data analysis, we analyzed the effects of ionospheric scintillation on GPS signals [2]. The ionospheric scintillation, caused by irregularities in the density of electrons present in the ionosphere, can weaken the signal received by the GPS receiver, causing degradation of positioning or even signal loss [3].…”

“…The DWT and NDWT were also applied in the S4 index time series [2]. The Figures (7) and (8) In the S4 index time series multiscale analysis, the NDWT also present better results than DWT.…”

“…As the NDWT is shift-invariant and represents a time series with the same number of coefficients at each scale, it is possible to detect the occurrence of hidden information such as stationary or non-stationary patterns and its time/frequency location. Recently, the NDWT has been applied in some areas of research: in GNSS signal analysis to investigate the ionospheric scintillation effect [2]; in daily temperature data to explore the time scale patterns in the relationship between the average ambient temperature and the number of deaths due to cardiovascular diseases [1]; in analysis of BSE and NSE indexes financial time series [5]; in waves pressure, height geopotential and thickness time series analysis in order to forecast rain events [10], among other applications. However, the use of NDWT should be much larger in signal analysis.…”