S ₄ index: Does it only measure ionospheric scintillation?

Abstract

The study of ionospheric scintillation has played a critical role in ionospheric research and also in satellite positioning. This is due to the growing influence of GNSS in navigation and remote sensing activities and also because the scintillation severely degrades the performance of this system. The main parameter that has been used to investigate the ionospheric scintillation impact on quality of GNSS satellite signals is the S ₄ index. However, we show that other effects such as multipath may mask the scintillation effects. Furthermore, we propose a method to separate the effect of multipath from scintillation in the S ₄ index by a non-decimated wavelet multiscale decomposition, which is shift invariant and more appropriate to use in the analysis of time series. The results show that the multipath effect is evident in the smoother scales of multiscale decomposition by wavelet in the weak scintillation period. Once identified and estimated, this effect can be removed from the S ₄ index series during strong scintillation periods and it becomes not significant in the scintillation S ₄ index wavelet analysis. Investigations using data from different stations and satellites demonstrate that the effect of ionospheric scintillation varies with station location and elevation angle of the satellite. Therefore, each case must be treated individually.

Appendix: NDWT pyramid algorithm

Given a time series {$X_{t} : t = 0, \ldots ,N - 1$}, the result of the filtering of $\{ X_{t} \}$ with the NDWT wavelet $\{ \tilde{h}_{k} \}$ and scaling $\{ \tilde{g}_{k} \}$ filters is given, respectively, by (Percival and Walden 2000)

$$\tilde{W}_{j,t} = \mathop \sum \limits_{k = 0}^{K - 1} \tilde{h}_{j,k} X_{t - k\bmod N}$$

(5)

$$\tilde{V}_{j,t} = \mathop \sum \limits_{k = 0}^{K - 1} \tilde{g}_{j,k} X_{t - k\bmod N} ,\quad t = 0,1, \ldots , N - 1$$

(6)

The two sequences given by (5) and (6), called NDWT wavelet and scaling coefficients, represent the NDWT of decomposition level $j$, and stipulate that the elements of $\tilde{W}_{j}$, $\tilde{V}_{j}$ and $\tilde{V}_{j - 1}$ are obtained by circularly filtering of $\{ X_{t} \}$ with the filters $\{ \tilde{g}_{j,k} \} , \{ \tilde{h}_{j,k} \}$ and $\{ \tilde{h}_{j - 1,k} \}$, respectively. The term “mod” from the equations refers to “maximal overlap discrete,” that treats time series X as if it were circular, making X to be represented with the same number N of coefficients at each scale.

However, it is possible to obtain $\tilde{W}_{j}$ and $\tilde{V}_{j}$ by filtering of $\tilde{V}_{j - 1}$ as in the following equation

$$\tilde{W}_{j,t} = \mathop \sum \limits_{k = 0}^{K - 1} \tilde{h}_{k} \tilde{V}_{{j - 1,t - 2^{j - 1} k\bmod N}}$$

(7)

$$\tilde{V}_{j,t} = \mathop \sum \limits_{k = 0}^{K - 1} \tilde{g}_{k} \tilde{V}_{{j - 1,t - 2^{j - 1} k\bmod N}} ,\quad t = 0,1, \ldots , N - 1$$

(8)

Equations (7) and (8) represent the NDWT pyramid algorithm, and they can be written in matrix form as

$$\tilde{W}_{j,t} = \tilde{B}_{j} \tilde{V}_{j - 1} \;{\text{and}}\;\tilde{V}_{j} = \tilde{A}_{j} \tilde{V}_{j - 1}$$

(9)

where the rows of $\tilde{B}_{j}$ contain circularly shifted versions of $\{ \tilde{h}_{k} \}$ after it has been upsampled to width $2^{j - 1} (K - 1) + 1$, that consists of inserting $2^{j - 1 }$ zeros between each of the K values of the original filter, and then periodized to length N, and with a similar construction for $\tilde{A}_{j}$ based upon $\{ \tilde{g}_{k} \}$(Percival and Walden 2000).

The NDWT also allows reconstructing $\tilde{V}_{j - 1}$ from $\tilde{W}_{j}$ and $\tilde{V}_{j}$. The inverse NDWT acts to restore the signal in the time domain from its decomposition, and it can be calculated via inverse pyramidal algorithm described by the following equation,

$$\tilde{V}_{j - 1,t} = \mathop \sum \limits_{k = 0}^{K - 1} \tilde{h}_{k} \tilde{W}_{{j,t + 2^{j - 1} k\bmod N}} + \mathop \sum \limits_{k = 0}^{K - 1} \tilde{g}_{k} \tilde{V}_{{j,t + 2^{j - 1} k\bmod N}} , \quad t = 0,1, \ldots , N - 1$$

(10)

which can be expressed in matrix notation as

$$\tilde{V}_{j - 1} = \tilde{B}_{j}^{T} \tilde{W}_{j} + \tilde{A}_{j}^{T} \tilde{V}_{j}$$

(11)

where $\tilde{B}_{j}^{T}$ and $\tilde{A}_{j}^{T}$ are transposed matrix of $\tilde{B}_{j}$ and $\tilde{A}_{j}$, respectively.