On the correlation between ROTI and S4

The correlation between Rate of TEC index (ROTI) and amplitude scintillation index S4 has been under focus for years, since both of the two are associated with ionospheric irregularities. Previous studies show that the behavior of the correlation between ROTI and S4 is not regular. To address this problem, in the present work the relation between S4, ROTI and the Differential Rate of TEC index (DROTI) has been investigated, assuming the single screening model for trans10 ionospheric radio propagation. The influences of the effective velocity and elevation angle and their variability on the correlation between ROTI and S4 are analyzed. Data from two stations of Scintillation Network Decision Aid (SCINDA) network are used for the study. Results show that the variability of effective velocity plays a crucial role on the correlation between the amplitude scintillation index and the TEC rate of change indices. Based on the Gaussian assumption of effective velocity distribution, a high correlation coefficient can be achieved only under low variability conditions. 15

scintillation spectrum and Fresnel scale (Li et al ,2007). A signal propagation model proposed by Du et al (2000) tried to relate ROTI with S4 with fixed effective velocity; the model was later again used by Amabayo et al (2015) to investigated the correlation between derived S4 from Du's model and the observed S4; their results show stronger correlations under some geographical locations. The limitation of Du's model as pointed out by Carrano et al (2019) is mainly the lack of consideration about detailed scintillation spectrum parameters; while it has been found that effective velocity is related to 35 both the velocity of ionospheric pierce point (IPP) and zonal velocity (Arruda et al, 2006;Chapagain et al, 2013). The assumption of constant effective velocity influences the correlation between ROTI and S4..
To further investigate this problem, the present work starts from the single layer phase screen model to derive the relationship between ROTI and S4; a formula of S4 is proposed in relation with the differential rate of TEC index (DROTI), the effective velocity and geometry. The correlation between ROTI and S4 are further studied, considering the randomity of 40 effective velocity. Section 2 gives the basic derivation of ROTI and the proposed methodology; Section 3 introduces the data used for experiments; Section 4 and 5 show the results and discussions; Finally,some conclusion are given.

ROTI calculation
TEC is derived by extracting ground GNSS observables from SCINDA files. For each slant path between satellite and 45 receiver on the ground, a slant TEC can be calculated assuming that the ionosphere is concentrated in the thin shell at a given height. To describe the disturbance of ionosphere TEC, rate of change of slant TEC is calculated. In addition, an index of the rate of slant TEC, which considers the average standard deviation of slant TEC rate has been proposed, and referred as ROTI by Pi et al (1997). Considering that the slant TEC is given by where +1 and are slant TEC at + 1 and time epochs; is time interval; usually the unit of ROT is TECU/min. ROTI has been defined by Pi et al (1997) as: where ⟨ ⟩ denotes averaging ROT during epochs. Experimentally, a threshold of 0.2 TECU/min is set to determine whether irregularity exists (Pi et al, 1997). The relation between ROTI and scintillation indices has already been studied to 55 verify if ROTI as a reliable indicator of ionosphere irregularity (Bhattacharyya et al, 2001;Yang and Liu, 2016). The index DROTI takes the standard deviation of DROT instead of , which is the second order TEC variation, given as Substituting with in equation (2), we get the expression of DROTI

Proposed method
In one-dimensional phase screen model, assuming that phase varies only along the x-direction, and according to the transport of intensity equation (Rino 1979), the following equation can be obtained: where is the unified signal amplitude intensity, 0 is its standard level under non-scintillation conditions. λ denotes the 65 signal wavelength. Further, the phase variation ( ) is related to ( ) along the signal path, according to the phase screening theory: = 2.8 × 10 −15 m is the radius of the electron. Spatial variations along x-direction are converted to temporal variations at the site of the receiver due to the relative motion of the irregularities with respect to the signal path. Then (5) was rewritten 70 as: The left side of (3) approximates to the variation of ROTI, represented as DROTI, and the right side approximates to the amplitude scintillation S4 (see Appendix). Therefore, from the above equation it can be written: where is the slant distance between phase screen ionosphere pierce point and the GNSS receiver.
denotes the radius of the earth, ℎ denotes the height of ionosphere pierce point from earth, usually considered 450km. ν is the effective velocity described in single phase screen model (Carrano et al, 2016;Rino, 1979). The velocity is related to ROTI variation, amplitude scintillation index S4 and scintillation intensity. To simplify the equation, it gives that

Correlation between ROTI and observed S4
The correlation between ROTI and observed S4, which was calculated from the observables is shown in Figure 4. To better 105 feature the correlation, a normalized ROTI was used by dividing all ROTI values with the maximum ROTI in the month (Yang and Liu. 2016). The correlation between ROTI and S4 looks very poor in this case, despite some stronger scintillation event existed in the same month. Most S4 index values are lower than 0.5, only ~20 cases indicate severe scintillation events, with S4 above 0.7. It should be noted that in some cases large S4 corresponded to small normalized ROTI; similarly, in some cases small S4 corresponded to large normalized ROTI. The overall correlation coefficient was 0.555 for Cape Verde and 110 0.566 for Dakar indicating a low correlation between ROTI and S4. The results indicate that large normalized ROTI with small S4 and similarly large S4 with very small ROTI can be found, showing that ionospheric scintillations are not directly related to ROTI.

the dependence of effective velocity
According to equation (9), S4 depends not only on ROTI but also on the effective velocity, leading to the assumption that the poor correlation between ROTI and S4 is due to the impact of the effective velocity. To validate this assumption, correlation 120 between observed ROTI and calculated S4 from equation (9) was implemented. To distinguish how the effective velocity influences the results, the variability of effective velocity in equation (8) is assumed to follow Gaussian distribution, with parameters [ , ]. The range of is estimated from equation (8) with available data, and set to be 50 m/s, indicating the mean value of effective velocity. The varies according to the ratio of ⁄ , indicates the standard deviation In the numerical experiments, ⁄ has a range from 0.1 to 3, the larger ⁄ is, the smaller variability of the effective velocity is 125 assumed, Figure 5 shows the correlation with different ⁄ values; it can be seen that the correlation coefficient decreases with decreasing ⁄ . The correlation coefficient is above 0.9 when ⁄ larger than 0.8; the correlation coefficient dropped drastically when ⁄ decreased from 0.6 to 0.1. The values are 0.78 when ⁄ = 0.4, 0.63 when ⁄ = 0.2, and 0.49 when ⁄ = 0.1. It appears that the correlation coefficient depends greatly on the randomness of the effective velocity. Figure 6 shows clearly the influences of ⁄ for both CVD and DKR stations, with the same μ = 50 set in Figure 5. The correlation 130 coefficient gradually decreased proportional to the ⁄ for both two stations. The geometric parameters including the satellite orbit, elevation and azimuth were derived in each calculation from the observables.

Figure 8: Correlation between observed ROTI and calculated S4 index by Equation (9) for one SCINDA stations in
September 2013; = m/s; the mean elevation angle varies from 30 to 90, with 5 as interval.

Discussion
The single-phase screening model proposed by Rino (1978) indicates that scintillation intensity was strongly related with the 165 scintillation spectrum and Fresnel scale. A series of successive studies further prove that the geometry factors are also crucial to decide the scintillation features (Carrano et al, 2016). Several indices have been used to study scintillation intensity, like S4, . Even ROTI is often used to investigate scintillation (Veettil and Aquino, 2017;Cherniak and Zakharenkova, 2018).
In this study, from equation (9), it is shown that S4 has better correlation with DROTI, rather than ROTI, giving a possible explanation of the different correlation coefficients between ROTI and S4 found in previous studies. From equation (9) it can 170 be seen that S4 is influenced not only by the irregularity itself represented by the TEC rate of change, but also by the effective velocity, the distance between satellite and receiver and also the propagation path. Carrano et al (2019) pointed out that correlation between ROTI and S4 is affected by the zonal velocity and zenith angle, and that only when zenith angle is less than 20 °can ROTI and S4 achieve a higher correlation. In this work, the randomity and variability of effective velocity and elevation angle are considered. Figure 5 shows that correlation between ROTI and S4 indeed varies according to the 175 variability of effective velocity. Figure 6 further shows the correlation degradation with decrement of ⁄ , which indicates the variability of the velocity under the Gaussian distribution assumption of effective velocity. Figure 7 confirms that it is the variability of effective velocity that influences the correlation results between ROTI and S4. Results in Figure 6  suggest that a high correlation between ROTI and S4 can be achieved when ⁄ > 1. When poor correlations between ROTI and S4 were found, a large variability of the effective velocity was the cause. The randomity and variability of 180 elevation angle shows very weak influence on the correlation coefficient as Figure 8 demonstrates. This doesn't mean that geometry has very minor impact on the correlation between ROTI and S4, since the effective velocity itself is dependent on geometry and can be strongly influenced by geometry factors. The single screen model proposed from equation (5) to equation (7) is in essence the same than the model described by while S4 it is related to the radio propagation conditions, showing how the signal will suffer from ionospheric irregular behavior (indicated by ROTI), but also from the effective velocity. The effective velocity is considered as combination of zonal drift velocity and the velocity of ionospheric pierce point (Arruda et al, 2006;Chapagain et al, 2013). Thus, in this study, a clear link between scintillation intensity and the effective velocity behavior has been established by equation (9).

Conclusion 195
In this work, the correlation between ROTI and S4 is investigated based on a derived mathematical model represented by equation (9) and under the assumption of single screening model for trans-ionospheric signal propagation. After taking the randomity and variability of effective velocity into account, this investigation reveals something new about the correlation between ROTI and S4 as follows.
(1) The correlation between ROTI and S4 is strongly affected by the variability of the effective velocity, and the 200 correlation coefficient is degraded when the variability increases. If the randomity of effective velocity is modeled as Gaussian distribution, a higher correlation coefficient is found when ⁄ > 1. At the same time, the value of effective velocity itself is independent with the correlation coefficient.
(2) The scintillation index S4 depends not onlyon the ionospheric irregularities, but also on the effective velocity and the trans-ionospheric radio propagation path. In fact, the effective velocity is also impacted by the geometry 205 factors and ionospheric status itself. S4 is closer linked with DROTI and can be derived when both DROTI and effective velocity are known.
(3) The physical meaning represented by S4 and ROTI are different. ROTI indicates the internal instability of ionosphere, and usually is considered as a proxy of ionospheric irregularity. S4 depends on the trans-ionospheric radio propagation conditions, and is defined by both of ionospheric irregularities and the effective velocity.