A Global Positioning System (GPS) scintillation channel model for the disturbed low-latitude ionosphere is developed by combining first-principles physics-based simulated low-latitude ionospheric irregularities together with an electromagnetic wave propagation model. Using this combined model, basic channel parameters, that is, phase variance, decorrelation length, decorrelation time, and coherence bandwidth, are computed and are favorably compared with recent ground-based low-latitude GPS experimental observations.
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 The ionosphere can have important effects on global navigation satellite systems, for example, the Global Positioning System (GPS). The primary ionospheric effects are group delay, giving absolute range errors, and amplitude and phase scintillation. Large-scale ionospheric structure, for example, the equatorial anomaly, can lead to group delay [Skone and Shrestha, 2002] with smaller-scale ionospheric irregularities generating scintillation effects [Su. Basu et al., 2001]. GPS scintillation occurs primarily at low and high latitudes but can also be observed at mid latitudes during magnetic storms. The largest GPS scintillation fading occurs at low latitudes.
 Many experimental studies have clearly demonstrated GPS amplitude and phase scintillations due to low-latitude ionospheric irregularities. However, a quantitative model for the GPS scintillation channel at low latitudes has not been developed. Such a model would be useful to develop techniques to mitigate against ionospheric scintillation effects on present and future global navigation satellite systems. In this study a channel model relevant to GPS scintillations in the low-latitude ionosphere is presented. In section 2 first-principles physics-based computer simulation model results for the Rayleigh-Taylor instability which causes equatorial ionospheric F region irregularities and associated GPS scintillation effects are presented. The spatial power spectrum of the ionospheric irregularities is presented. In section 3 a low-latitude GPS scintillation channel model is developed. In section 4, using the model, basic channel parameters, that is, phase variance, decorrelation lengths, decorrelation times, and coherence bandwidths, are computed and compared with recent ground-based GPS observations. Finally, in section 5 we summarize and discuss our results.
2. Low-Latitude Ionospheric F Region Irregularities
Figure 1, which is descriptive of the low-latitude F region ionosphere, shows the geometry. Recently, a three-dimensional (3-D) time-dependent computer simulation model for the Rayleigh-Taylor instability [Keskinen et al., 2003], which is the causative mechanism for GPS scintillations in the low-latitude ionosphere, has been developed. Figure 2 gives an example of the nonlinear spatial evolution of the ionospheric electron density in the Rayleigh-Taylor bubbles in the xz plane at t = 1 hour after triggering. The initial conditions used to trigger the bubbles in Figure 2 are of the form (δne/ne)f(y)cos(kxx − ωt) with δne/ne = 0.04. The east-west perturbation wavelengths kx−1 ranged from 200 km down to approximately 20 km. A total of fifteen modes were initially seeded. The spatial resolution in Figure 2 is 1 km in the vertical, north-south, and east-west directions. Large depletions in ionospheric electron density can be seen up to high altitudes.
Figure 3 shows a different simulation of the nonlinear evolution of equatorial ionospheric bubbles but at higher resolution of 20 m and completely random seeding. The maximum amplitude of the initial relative electron density perturbation is 0.04. The span of ionospheric irregularity scale sizes in Figure 3 are at, above, and below the Fresnel-scale size Lf where the Fresnel size is defined Lf = (λD)1/2 with λ the wavelength and D the distance from the scatterer, that is, ionospheric irregularity region, to the receiver. The primary scale size regime leading to scintillations in a ground-based receiver is at, above, and below the Fresnel size [Sreenivasiah et al., 1976; Ishimaru, 1978; Tatarski, 1971; Yeh and Liu, 1977; Knepp and Wittwer, 1984]. For GPS(L1) λ = 19 cm with GPS(L2) λ = 25 cm. For GPS(L1) we have Lf ≃ 300 m for D = 400 km. Figures 4 and 5 gives the typical power spectra P(kx) and P(kz) computed from the 3-D high-resolution code. From a series of simulations of the Rayleigh-Taylor instability in the Fresnel scale size regime, we have found that the 3-D spatial power spectrum of the electron density can be written approximately
with n ≃ 3.6–4.6, Lx ≃ 0.5–2 km, Ly ≃ 80–100 km, Lz ≃ 3–10 km. This power spectrum is consistent with experimental observations.
3. GPS Scintillation Channel Model
 We consider a monochromatic spherical wave of the form
with k2 = (ω2/c2)(1 − ωpe2/ω2), ωpe2 = 4πnee2/me, and ρ is a position vector in (x,y) plane transverse to the propagation direction z. The wave is generated from a transmitter a distance zt from the ionosphere with a receiver a distance zr below the ionosphere, for transionospheric propagation, as shown in Figure 6. θ = 0 in Figure 1 is assumed. The function U obeys the parabolic wave equation [Tatarski, 1971; Yeh and Liu, 1977; Ishimaru, 1978]
 For GPS applications, instead of a monochromatic wave, one must consider a wide bandwidth signal which we write as m(t)exp(iω0t) with the carrier frequency (L1/L2) denoted by ω0 and the modulation m(t). For GPS the modulation frequency spectrum is of the form m(ω) = Tc(sin πωTc/πωTc)2 with Tc the chipping period. For the P code the chipping period is the reciprocal of 10.23 MHz while for the C/A code it is 1.023 MHz. The complex envelope at the receiver can be written
Furthermore, it can be shown [Ishimaru, 1978] that the mean received power at the receiver can be written as
where Γ is the mutual coherence function given by Γ = 〈U(ρ1, z1, ω1, t1)U*(ρ2, z2, ω2, t2)〉.
 The mutual coherence function is found by solving equation (3) given a representation of the 3-D power spectrum of the ionospheric electron density irregularities. A solution for Γ, using equation (1) with n = 4.5 and previous methods [Knepp, 1983], can be written
where f1 = 1 + iωg1/ωcoh, f2 = 1 + iωg2/ωcoh, g1 = (2lx4/(lx4 + ly4))1/2, g2 = (2ly4/(lx4 + ly4))1/2, and = ω0/σϕ. Here Γ has been normalized by the exact free space value.
 The phase variance from this model can be written
with re = e2/mec2, λ is the wavelength, L is the layer thickness, and is the variance of the ionospheric electron density. The decorrelation lengths in the x and y directions can be written
For distances greater than the decorrelation length, for example, x ≫ lx the mutual coherence function departs from the free space value. The coherence bandwidth can be written
For frequency differences greater than the coherence bandwidth, that is, ω ≫ ωcoh, the mutual coherence function is distorted from its free space value. The decorrelation time is
where T0 is a characteristic time for the ionospheric irregularities.
 The model discussed in Section 3 is applied to low-latitude GPS scintillations. Figure 7 gives representative values, using equation (9), for the decorrelation length in the x direction lx for a range of the spatial spectrum outer scale Lx. For the phase standard deviation σϕ we take consider GPS L1. For GPS L2 we have σϕ approximately (25/19) = 1.3 times larger. The σϕ as computed from the model are consistent with observations [Su. Basu et al., 2001; S. Basu et al., 2001; Aarons et al., 1997]. Figure 7 indicates that the decorrelation lengths in the x direction lx are on the order of a kilometer depending on σϕ. Here zt = 20,200 km and zr = 400 km have been used. These decorrelation lengths are in agreement with the east-west decorrelation sizes on the order of a kilometer for GPS L1 scintillation measured recently [Kintner et al., 2004] in the low-latitude ionosphere. Using equation (10), Figure 8 gives the decorrelation lengths in the y direction along the geomagnetic field direction and are larger than perpendicular x direction lengths due to the larger outer scales along the geomagnetic field. Kintner et al.  also showed that the low-latitude north-south decorrelation lengths were larger than the east-west lengths. The magnitude of the north-south decorrelation lengths could not be determined exactly by Kintner et al.  but were found to exceed approximately a few kilometers.
Figure 9 gives the decorrelation time computed from the model and is on the order of seconds. Here we have taken T0 = Lx/Vx where Vx is the ionospheric drift with respect to the receiver. These decorrelation times are consistent with those recently measured by Kintner et al.  on the order of seconds for GPS L1 using similar drifts. Using equation (11), Figure 10 gives the coherence bandwidth for GPS L1 and ranges from tens to hundreds of MHz depending on the phase standard deviation and other parameters. The coherence bandwidth values for GPS L2 can be found by scaling arguments and are (19/25)4 = 0.33 times smaller.
5. Summary and Discussion
 A quantitative GPS scintillation channel model for the disturbed low-latitude ionosphere has been developed. This has been accomplished by combining a time-dependent first-principles computer simulation model of the low-latitude ionospheric irregularities with a electromagnetic wave propagation model. Using this approach we have computed basic channel model parameters, that is, phase variance, decorrelation lengths, decorrelation times, and the coherence bandwidth. The phase variance is found to be proportional to the outer scale of the power spectrum for scale sizes at and near the Fresnel size. In addition, the phase variance scales directly with the ionospheric electron density variance. The decorrelation lengths transverse plane to propagation directly scale with the spectrum outer scale and inversely with the phase standard deviation. The coherence bandwidth follows a scaling with the fourth power of the signal frequency and directly proportional to the outer scale. The decorrelation lengths and decorrelation times from the model are consistent with recent ground-based GPS scintillation observations in the low-latitude ionosphere.
 The output of this model could be used to provide a predictive physics-based scintillation model for GPS software receivers. Carrier phase tracking loops are susceptible to scintillation. The signal to noise threshold for reliable tracking is strongly dependent on scintillation levels. Most GPS software receiver models use empirical models for phase variance. The physics-based model developed in this study can provide a predictive capability for the phase variance and other channel parameters, that is, decorrelation times, decorrelation lengths, and coherence bandwidth, not provided by empirical models.