Statistical Uncertainty in the Frequency Dependence of the Intensity Scintillation Index (S-4)

Abstract

The intensity scintillation index (S-4) exhibits a linear frequency dependence derived from a power law phase screen model, with the exponent determined by the phase spectral index (p). However, it is well known that a departure from linearity occurs as S-4 increases and saturates under strong scatter conditions. Additionally, statistical errors over a finite time window can also cause the scintillation statistic to deviate from the power law dependence on frequency. This paper presents a parametric analysis on the deviation in the S-4 frequency dependence from the power law form. This investigation focuses on the contribution of statistical uncertainty in S-4 calculations to the deviation in the frequency dependence according to the selected parameters: S-4 and Fresnel frequency (f(F)). The deviation is determined by comparing p estimates calculated from the power law relationship of S-4 at different L-band frequencies with those inferred from a model-fitting method using a multi-frequency global navigation satellite system scintillation data set. The results show the well-known departure from linearity as S-4 increases, and reveal increasing deviations with decreasing f(F). The spectral analysis indicates that the variability at the lowest frequencies of intensity spectra contributes to the statistical errors in S-4 calculations due to the finite time window, particularly when scintillations are dominated by low-frequency contributions at low f(F). Simulation results of scintillation realizations show much lower deviations with a 30-min compared to 1-min window. This suggests a 1-min window can be too short for reliable S-4 calculations due to statistical uncertainty, especially at low f(F). Plain Language Summary The S-4 index is a widely used measure to quantify scintillation strength. The relationship between S-4 values at different signal frequencies is theoretically expected to follow a linearity. However, this simple relationship is only valid under certain conditions, and the calculation of S-4 with a finite time window can introduce uncertainty that causes deviations from the expected frequency dependence. In this study, we analyze the uncertainty in the relationship between S-4 values at different frequencies, considering the specific scintillation conditions. Our observations confirm two main findings: (a) a well-known departure from linearity at high levels of scintillation, and (b) deviations caused by statistical uncertainty due to large-scale fluctuations over a finite time window for S-4 calculations. We validate these findings through simulations that cover a wide range of scintillation scenarios.