### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Extended Scintillation Propagation Model for Low-Latitude Bubbles
- 3. Effective Model of a Bubble
- 4. Numerical Modeling Calculations and Comparison With Real Data
- 5. Conclusions
- Acknowledgments
- References

[1] A previously developed scintillation propagation model for *L* band signals on transionospheric paths has been further extended to describe the effects caused by the localized structure of plasma bubbles in the low-latitude ionosphere. This takes into account quasi-deterministic and random structures typical of bubbles. The model can produce signal statistical moments (power spectra, correlation functions, scintillation index, etc.) and generate random time series including the case of through bubble propagation. The simulated random time series of the field demonstrate the characteristic nonstationary behavior caused by the presence and motion of the bubble structures through the path of propagation, showing that strong enhancements of the scintillation index (S_{4}) can occur depending on the parameters of the bubble and the path. Modeling results are compared with scintillation records due to bubbles passing through GPS signal paths to a receiver at Douala, Cameroon. This shows good agreement providing validation for the bubble and propagation model.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Extended Scintillation Propagation Model for Low-Latitude Bubbles
- 3. Effective Model of a Bubble
- 4. Numerical Modeling Calculations and Comparison With Real Data
- 5. Conclusions
- Acknowledgments
- References

[2] Fluctuations of the electron density of the ionosphere result in the scintillation phenomenon, which may adversely affect navigation and other Earth-satellite telecommunication links. In extreme cases scintillation of a signal may even result in loss of the phase lock and, if this occurs on a sufficient number of satellites, positioning may no longer be possible. A special case of such scintillation effects is when midscale inhomogeneities (of scale of the order of hundreds of km) occur in the ionosphere. Patches, arches, or blobs in the high-latitude ionosphere, or bubbles in the low latitudes may cause strong enhancement of the level of scintillation, characterized by the values of scintillation index S_{4}, which may reach the range of values corresponding to strong scintillation conditions. The St. Petersburg-Leeds scintillation propagation model, [*Gherm et al.*, 2000, 2005a] developed initially for the case of smoothly inhomogeneous background ionosphere (not containing midscale local inhomogeneities such as low-latitude plasma bubbles), is particularly useful for describing these situations, including the case of strong scintillation with values of the fractional electron density up to 100% for operating frequencies of 1GHz and above. This turns out to be possible because, at this range of frequencies, the regime of strong scintillation is never likely to occur inside the ionospheric layer. Therefore, the perturbation theory can be employed for describing the propagation in the ionosphere, and then the field immediately below the ionosphere can be determined. Since strong scintillation may well occur for the field when propagating below the ionosphere, to convey the field at the level immediately below the ionosphere down to the Earth's surface, the general theory of the random screen is employed. We term this the hybrid scintillation propagation model.

[3] The hybrid model has now been further extended to enable description of the effects due to local midscale (mesoscale) ionospheric inhomogeneities, e.g., bubbles of the low-latitude ionosphere or patches at high latitudes. The effect of patches in the high-latitude or polar ionosphere has been previously described by *Maurits et al.* [2008]; the effect of bubbles in the low-latitude ionosphere will be treated here. The scintillation effects for bubbles in the low-latitude ionosphere differ from the effects resulting from turbulence in the midlatitude ionosphere due to the occurrence of the moving midscale local inhomogeneities that make the stochastic signal essentially statistically nonhomogeneous. Consequently the hybrid propagation model had to be significantly modified to account for the effects in these statistically nonstationary conditions. This extension will be outlined below and then examples of modeling the signal scintillation effects due to the bubbles will be presented.

### 2. Extended Scintillation Propagation Model for Low-Latitude Bubbles

- Top of page
- Abstract
- 1. Introduction
- 2. Extended Scintillation Propagation Model for Low-Latitude Bubbles
- 3. Effective Model of a Bubble
- 4. Numerical Modeling Calculations and Comparison With Real Data
- 5. Conclusions
- Acknowledgments
- References

[4] To make this paper more self contained, the previously most recent version of the scintillation propagation model enabling the description of the statistically nonstationary effects due to time varying midscale (bubble-like ionospheric) inhomogeneities will be briefly outlined below. Addressing readers to *Gherm et al.* [2005a] and *Maurits et al.* [2008] for details, the propagation in the inhomogeneous ionospheric layer is described within the complex phase method (CPM), which is the extension of the classic Rytov's approximation to the case of a point source field in a smoothly inhomogeneous medium [*Gherm et al.*, 2005b]. The field in the fluctuating transionospheric channel is represented as follows:

Here *r* and *t* are the space and time coordinates of the point of observation, *ω* is the angular frequency. *E*_{0} is the field when there are no fluctuations of the electron density of the ionosphere that is normally described in the geometrical optics approximation. Function *R*(r, *ω*, *t*) in equation (1) is a random factor (random phasor) taking account of fluctuations of the electron density in the ionosphere. Using the CPM technique, it is treated in terms of the complex phase

Auto- and cross-correlation functions of the phase *S* and log amplitude *χ*, necessary for generating the field in and at the bottom of the ionosphere can be conveniently to expressed through their spatial spectra, which [*Gherm et al.*, 2005a; *Maurits et al.*, 2008], are, respectively, given by

[5] Function *B*_{ɛ}(*s*, κ_{s}, κ_{n}, κ_{τ}) in equations (3)–(5) is the spatial spectrum of fluctuations of the dielectric permittivity of the ionosphere due to the electron density fluctuations. It depends on variable *s* along the path of propagation, which is not necessarily a straight line. This dependence on *s* is defined by a product of factors (ɛ_{0}(*s*) − 1) and *σ*_{n}^{2}(*s*), where function ɛ_{0}(*s*) is the dielectric permittivity of the 3-D background ionosphere along the propagation path and *σ*_{n}^{2}(*s*) is the variance of the relative electron density fluctuations. In turn, function (ɛ_{0}(*s*) − 1) is proportional to the squared plasma frequency in the background ionosphere, *ω*_{pl}^{2}(*s*), and thus the distribution of the electron density along the propagation path in the background ionosphere *N*(*s*). Function *B*_{ɛ} in (1)–(3) is taken at κ_{s} = 0. Points (0, *S*_{0}) are the start and end points of the path of propagation. Local spectral variables are κ_{s} along the path and κ_{n} and κ_{τ} in the plane perpendicular to the path of propagation at each point. Constant *k* is the vacuum wave number. Elements D_{ab} in (3)–(5) are linked with the curvature of the phase surface of the propagating wave [*Gherm et al.*, 2005b].

[6] Finally, function *J*(*s*) in equations (3)–(5) is the Jacobian of transformation of the transversal spectral variables κ associated with the point of observation *s* into current transversal spectral variables along the reference ray in the ray-centered coordinate system. This is performed according to the following equations:

Again, the details relevant to equations (3)–(6), are given by *Gherm et al.* [2005a]. In turn, variances of the field phase and log-amplitude fluctuations are derived from equations (3) and (4) by integrating with respect to the transversal wave numbers.

[7] The model of the spatial spectrum *B*_{ɛ}(*s*, κ_{s}, κ_{n}, κ_{τ}) is chosen as a single-slope inverse power law of the following form:

The latter 3-D model is introduced to account for fully 3-D local random inhomogeneities with three different outer scales according to the relationship *K*_{ɛ} = 2*πl*_{ɛ}^{−1}, where *l*_{ɛ} are the three different outer scale lengths. These three different outer scales of the ionospheric turbulence were introduced by *Maurits et al.* [2008] to adequately model the electron density fluctuations, e.g., in the polar ionosphere [see, e.g., *Tereshchenko et al.*, 2005]. Here, to model the effects due to low-latitude bubbles, the two cross-magnetic field outer scales are chosen to be equal. Wave number κ_{tg} is the spectral variable along the Earth's magnetic field, and κ_{tr1} and κ_{tr2} are the two cross-magnetic field spectral variables. Additionally, the dependence of the variance of fluctuations of the fractional electron density with distance along the reference ray is also taken into account in the model in (7). Parameter *p* is the spectral index; *C*_{N}^{2} is the normalization coefficient; ɛ_{0}(*s*) is 3-D distribution of the dielectric permittivity of the background ionosphere along the reference ray, and *σ*_{N}^{2}(*s*) is the variance of the fractional electron density fluctuations.

[8] In the scope of the CPM, the normal distribution of *S* and *χ* follows from the central limit theorem, which is used in the following procedures for generating random series of the field. The presence of the ionospheric mesoscale structures makes the field statistically nonstationary, so that functions (3)–(5) are really the functions of two positions (not only the difference variable), or two different times, if the approximation of the “frozen drift” is locally introduced. The procedure of generating the random nonstationary series of the field on a certain surface properly chosen below the ionosphere is based on the Cholesky decomposition (factorization) of matrices (representing the two-point correlation functions) into the product of the low triangular matrix and its transpose and on the subsequent multiplication of the upper triangular matrix by a vector whose elements are noncorrelated Gaussian random numbers with mean zero and variance one [*Devroye*, 1986]. In so doing, the random realizations of the phase and log amplitude are generated, which form the nonhomogeneous random screen below the ionosphere. It is worthwhile stressing that, by contrast with the widely known multiple-phase screen technique [*Knepp*, 1983], which is employed to construct numerical solutions to stochastic problems making use of different types of the effective phase screens [*Grimault*, 1998; *Caouren*, 2002; *Beniguel*, 2002], here it is not an effective phase screen that is employed but a real physical random field is constructed, calculated for a conveniently chosen surface below the ionosphere. It is further conveyed down to the Earth's surface by a standard random (not necessarily phase) screen technique.

### 3. Effective Model of a Bubble

- Top of page
- Abstract
- 1. Introduction
- 2. Extended Scintillation Propagation Model for Low-Latitude Bubbles
- 3. Effective Model of a Bubble
- 4. Numerical Modeling Calculations and Comparison With Real Data
- 5. Conclusions
- Acknowledgments
- References

[9] Without the availability of a suitable concise accurate physical model of a bubble, an effective model of the bubble is utilized employing the following normalized basic function [*Maurits et al.*, 2008]:

Here (*x*_{2} − *x*_{1}) is the scale of the inhomogeneity, *a*_{1} and *a*_{2} are its “wall” thickness. By means of (8) a 3-D structure can be modeled as the product

with three different scales and different thickness of the walls. Function (9) is used “to embed” a bubble into the background ionosphere, so that any chosen background model *Ne*_{0}(**r**) is modulated by (9) as follows:

Here the coefficient A is a constant, which is negative for the depletion of electron density that occurs in a bubble. The movement of the bubble is accounted for by the dependences of **r**_{1} and **r**_{2} on time in the approximation of the “frozen drift.” The thickness of the walls is given by the parameters **d**_{1}, **d**_{2}.

[10] The model (8)–(10) was used to modulate the background large-scale ionosphere. The NeQuick model [*Radicella and Leitinger*, 2001] was employed to characterize the low-latitude background ionosphere, the parameters of which were chosen for a particular geographic location and time. For the model of the electron density fluctuations, a single-slope inverse power law spatial spectrum of the random inhomogeneities (7) was utilized for the numerical modeling of the effects of scintillation. Additionally, the standard deviation of the plasma electron density *σ*_{0}(**r**, *t*) was also modified for the area inside the bubble in the following way:

B is an appropriately chosen positive constant, parameters **d**_{1} and **d**_{2} in (10) and (11) should be taken to be slightly different values to result in the deviations of the plasma density being maximal in the walls of the bubble, *σ*_{N}(**r**, *t*) is the same as in equation (7) for the spatial spectrum of the electron density fluctuations. The models for the electron density depletion in a single bubble (10) and the standard deviation of the electron density outside and inside the bubble, modeled by (11), are illustrated in Figure 1 (left) and Figure 1 (right), respectively. In particular, Figure 1 (right) clearly shows the maximal values of the electron density deviation in the walls of the bubble, which corresponds to the common conception of a bubble structure. In the following consideration a single- or double-bubble propagation scintillation model is used, as appropriate, to numerically model the scintillation effects. We reiterate here that this model of a bubble is not one based on the physical principles of plasma dynamics and thus is not able of replicate the detailed structure of local plasma irregularities in actual bubbles. Rather, its aim is to adequately model and simulate the effects of transionospheric signals passing through low-latitude ionosphere plasma bubbles.

### 4. Numerical Modeling Calculations and Comparison With Real Data

- Top of page
- Abstract
- 1. Introduction
- 2. Extended Scintillation Propagation Model for Low-Latitude Bubbles
- 3. Effective Model of a Bubble
- 4. Numerical Modeling Calculations and Comparison With Real Data
- 5. Conclusions
- Acknowledgments
- References

[11] In this section results of modeling the effect of the passage of a bubble through propagation paths from GPS satellites to a ground receiver are presented. The receiver is considered to be located at Douala, Cameroon (4°0′ N, 9°42′ E) as experimental data obtained at Douala was kindly provided courtesy of Dr. B. Arbesser-Rastburg of ESA-ESTEC. Below the results of modeling the two different cases are presented.

[12] First, a signal from the geostationary satellite PRN131 was considered which has an azimuth of 92.88° and an elevation of 27.97°at the receiving location of Douala. The background ionosphere without the bubble was modeled using NeQuick for 9 June 2004 and for an F10.7 flux of 100 corresponding to the time and conditions of the experimental data with which the simulation was to be compared. Figure 2 shows the time dependence of the electron density of the ionosphere along the line of sight for a bubble passing through the path from the satellite to a ground-based receiver; Figure 2 (left) shows the depleted electron density along the slant path (the vertical axis is the distance along the raypath from the receiver) and Figure 2 (right) the corresponding time dependence of the slant total electron content (TEC) in TEC units. The ionosphere clearly shows a depleted area as the bubble passes through the propagation path. Figure 3 shows a time series of the amplitude variation at the receiver due to the passage of the bubble. Focusing and defocusing because of the large-scale structure of the bubble are observed. The corresponding time series of the phase can be seen in Figure 4 where Figure 4 (left) shows the total phase variation, dominated by that caused by the electron density depletion associated with the bubble structure while Figure 4 (right), by means of a high-pass filter, shows the detrended phase and thus just the stochastic part of the variation resulting from the small-scale time-varying irregularities. Figure 5 (left) shows actual measurements of scintillation of the signal of the geostationary satellite PRN131 at Douala obtained on 9 June 2004. Results of simulation for the same location and same time period, obtained according to the modeling technique described above, are shown in Figure 5 (right) also as the dependence of the scintillation index S_{4} as a bubble passes through the propagation path from the geostationary satellite. It is clear that there is very good agreement between the experimental results and those of the simulation in the framework of a single-bubble scintillation propagation model. To provide this good agreement the following parameters of a bubble were introduced: horizontal (west-east) scale of 150 km, a vertical scale of 400 km and a very large south-north scale. The effective velocity was just defined by the west-east drift velocity of the bubble and was 150 m/s as the satellite velocity would not contribute appreciably to the effective (relative) velocity of the bubble through the path. In the bubble the depletion level was taken as −90% and parameter B in equation (11) for the standard deviation of plasma density in a bubble was 9, which resulted in a tenfold enhancement with respect to the deviations of the density outside the bubble. The other parameters of the random inhomogeneities in the bubble were: a spectral index of 3.7, a cross-magnetic outer scale of 4 km and an aspect ratio of 10.

[13] A series of figures for the next example illustrate the case where the single bubble model is clearly not adequate. In these figures the experimental results and the results of simulation are presented for two satellites (PRN26 and PRN29) of the GPS constellation visible at Douala on 7 June 2004 in the period of time from 1700 to 2400 UT. In Figure 6 (left) shows the records of the scintillation index S_{4} against time for the signals from satellites PRN26 (blue) and PRN29 (green) and Figure 6 (right) presents the trajectories of these satellites in elevation and azimuth as seen from the receiver at Douala. Presence of multiple peaks of the scintillation index can most likely be attributed to a number of traveling eastward bubbles. Introducing the two-bubble scintillation propagation model, the scintillation events occurring in the time interval between 2030 and 2230 UT are modeled. For the time of approximately 21:00 the elevation angles of the satellites are 75° and 80°. The same two events for the period mentioned are shown in Figure 7 (left) on a larger time scale. The results of modeling these events in the framework of the double-bubble scintillation propagation model are shown in Figure 7 (right). As can be seen, there is good agreement between these results (Figure 7 (right)) and the experimental data (Figure 7 (left)) for S_{4} values, for the start and finish times and relative delays of the scintillation patterns for the two satellites. This is true for both of the two peaks of scintillation activity, each of about an hour's duration centered, around 21:2 and 21:75 h.

[14] To provide the best fit between the experimental results and the results of modeling for this case, the geometrical parameters of the first (in time) bubble were the same as in previous case. For the second bubble the horizontal scale was 90 km. The depletion level was −20% for both the bubbles, and the distance from one bubble to another was 200 km. The width of the wall was slightly altered from the value taken for the first bubble to provide the best possible match to the shape of the second scintillation peak shown in Figure 7 (right). The effective velocity was just defined by the west-east drift velocity of the bubbles and was 100 m/s as the satellites velocities did not contribute to the effective (relative) velocity of the bubble through the path as the movement of the satellites chosen for the scintillation modeling was in north-south direction (i.e., perpendicular to the drift velocity of the bubble). All the characteristics of random inhomogeneities in both the bubbles were the same as in the first case of the single bubble model.

[15] For simulation of the background ionosphere, in the both cases the NeQuick profiles were chosen for the location near the point of observation and dates of the respective measurement data. The transmission frequency used was the GPS L1, i.e., 1575 MHz.

[16] Finally, the time correlation function of the signal intensity for one of the satellites PRN26, PRN29 was determined. The time series of the full field generated for the period of 2030 to 2230 UT (constructed from the amplitude and phase time series similar to those shown in Figures 3 and 4 for the geostationary satellite PRN131) was divided into six intervals to produce the field intensity time correlation function for each of these intervals. These six correlation functions are shown in Figure 8. As can be seen, there is no significant change in the time correlation radius at the value of 0.5, which is estimated to be slightly less than 1 s. Only small changes in the correlation can be observed toward the extremities of the curves. The absence of a major difference in the time correlation radius of the field fluctuations for time intervals inside and outside the time of bubble passage though the propagation path can be explained by the fact that, in the particular case considered, the regime of weak scintillation always occurred with the level of scintillation index S_{4} never exceeding 0.3.