By continuing to browse this site you agree to us using cookies as described in About Cookies
Notice: Wiley Online Library will be unavailable on Saturday 7th Oct from 03.00 EDT / 08:00 BST / 12:30 IST / 15.00 SGT to 08.00 EDT / 13.00 BST / 17:30 IST / 20.00 SGT and Sunday 8th Oct from 03.00 EDT / 08:00 BST / 12:30 IST / 15.00 SGT to 06.00 EDT / 11.00 BST / 15:30 IST / 18.00 SGT for essential maintenance. Apologies for the inconvenience.
 It can be important to determine the correlation of different frequency signals in L band that have followed transionospheric paths. In the future, both GPS and the new Galileo satellite system will broadcast three frequencies enabling more advanced three frequency correction schemes so that knowledge of correlations of different frequency pairs for scintillation conditions is desirable. Even at present, it would be helpful to know how dual-frequency Global Navigation Satellite Systems positioning can be affected by lack of correlation between the L1 and L2 signals. To treat this problem of signal correlation for the case of strong scintillation, a previously constructed simulator program, based on the hybrid method, has been further modified to simulate the fields for both frequencies on the ground, taking account of their cross correlation. Then, the errors in the two-frequency range finding method caused by scintillation have been estimated for particular ionospheric conditions and for a realistic fully three-dimensional model of the ionospheric turbulence. The results which are presented for five different frequency pairs (L1/L2, L1/L3, L1/L5, L2/L3, and L2/L5) show the dependence of diffractional errors on the scintillation index S4 and that the errors diverge from a linear relationship, the stronger are scintillation effects, and may reach up to ten centimeters, or more. The correlation of the phases at spaced frequencies has also been studied and found that the correlation coefficients for different pairs of frequencies depend on the procedure of phase retrieval, and reduce slowly as both the variance of the electron density fluctuations and cycle slips increase.
 For various reasons it is of importance to determine how well the signals of different frequencies in L band are correlated. For example, in the future both GPS and the new Galileo satellite system will broadcast three frequencies enabling more advanced three frequency correction systems so that a future challenge is to incorporate multifrequency, in-the-clear transmissions. Another challenge is the optimum way to decode signals from the present L2 signal robustly without access to the encryption algorithm. State-of-the-art techniques for this rely on cross correlation of the signals received at L1 and L2 to improve the likelihood of acquiring and maintaining lock on the L2 signal. Thus it is important to determine how well these signals are correlated. Lack of correlation will also increase range error in dual-frequency correction using L1 and L2 even in the case of perfect decoding. A significant factor in reducing this correlation would be scintillation of the signals resulting from propagation through time-varying small-scale irregularities in the ionosphere, specifically, the effects due to diffraction on local random inhomogeneities of the ionosphere. It has been found that GPS (or other GNSS) receivers can suffer from cycle slip or even phase lock loss in conditions of severe scintillation. Such conditions are especially likely at polar, high (auroral) and low latitudes (±30 of the geomagnetic equator) where scintillation effects are the most severe. Thus it is vitally important to consider the effects of the different phase decorrelation and additional range error in dual-frequency ionospheric correction specifically due to the field diffraction on random inhomogeneities of the ionosphere of the spatial scales smaller than the appropriate main Fresnel zone size.
 Thus this paper addresses the effect of electron density irregularites on transionopsheric signals and in particular the effect of the scintillation on dual-frequency GNSS positioning resulting from diffraction of the L band signal field on the small-scale local random inhomogeneities. In particular, the contribution of the diffraction to the range error in the dual-frequency measurements and the correlation of the phases of the fields at different frequencies are determined for a range of scintillation severity.
 Recently Kim and Tinin [2007, 2009a, 2009b] have published work treating the effects of fluctuations of electron density of the ionosphere on the accuracy of the GNSS range measurements, including diffraction effects for dual-frequency operation. In the work presented by Kim and Tinin , this was done by means of perturbation theory in the geometrical optics (GO) approximation for an isotropic ionosphere with large-scale structure. The theory by Kim and Tinin [2009a, 2009b] was further developed to describe both the effects of large-scale ionospheric structures and small-scale random electron density fluctuations within the range of the Rytov's approximation, taking account of the first and second approximations of the method. In these later papers, however, the authors also concluded that use of Rytov's first-order approximation was satisfactory to describe the effects of weak scintillation due to the small-scale random inhomogeneities. The first and second Rytov's approximations were employed to derive the representations for different order ionospheric errors due to the large-scale background ionosphere. However, as far as the large-scale inhomogeneities are concerned, all these results reduce to what can easily be obtained making use of the GO approximation. Moreover, within the GO approximation, this can be done for the more realistic case of the field from a point source and inhomogeneous ionosphere [Kim and Tinin, 2007] and further also for an anisotropic ionosphere [Gherm et al., 2006a], rather than for a plane wave and homogeneous background ionosphere as shown by Kim and Tinin [2009a, 2009b]. Additionally, a much more precise solution to the ray problem than that provided by the Rytov's approximation is required when treating high-accuracy GNSS range measurements [Gherm et al., 2006a] in an anisotropic ionosphere. To study diffraction effect contributions, modification of our previously developed scintillation propagation model [Gherm et al., 2000, 2005; Maurits et al., 2008; Zernov et al., 2009], which enables the effect of electron density fluctuations on transionospheric raypaths at L band frequencies to be determined even for the case of strong scintillations, is required. This will be done in the present work.
 In the work by Gherm et al.  the propagation model for signal fluctuations in the transionospheric fluctuating channel was developed based on Rytov's approximation and, in particular, the effects of fluctuations of the ionospheric electron density on the accuracy of the range measurements was also investigated. In this paper, in contrast to the work by Kim and Tinin [2009a, 2009b], a more general version of Rytov's approximation (the complex phase approximation) has been developed in order to consider the effects of local ionospheric inhomogeneities, including the diffraction effects on the inhomogeneities with the spatial scales smaller than the Fresnel zone size. This takes account of the inhomogeneous background medium and the field point source utilizing ray-centered variables. Calculations have been made and are presented in this paper which assesses the additional range errors (including the error due to diffraction effects on local random ionospheric inhomogeneities) for different conditions of propagation. Rytov's approximation, as is well known, is only valid for weak scintillation. Our further investigation, specifically for L band frequency propagation on transionospheric paths, has shown that this cannot be comprehensively described in the scope of Rytov's approximation. This is especially true when propagation in the high-latitude and polar ionosphere conditions on fairly oblique paths is considered. In these cases the conditions of strong scintillations may often occur, so that another method, valid for strong scintillation conditions, should be employed.
 To enable consideration of the case of strong scintillation in L band, the hybrid method was developed [Gherm et al., 2005], the validity of which is not restricted to the case of the weak scintillation approximation, but is also capable of treating the strong scintillation of the field on transionospheric paths of propagation. It is particularly important to study such severe conditions which can result in loss of lock in GPS receivers and thus considerable degradation in positioning accuracy. Routine calculations showed that, within the ionospheric layer, the regime of strong scintillation is not normally encountered but may occur on the Earth's surface after propagation below the ionosphere down to the Earth. Therefore, in the hybrid method, propagation within the ionospheric layer, where electron density fluctuations are occurring, is described using Rytov's approximation, whereas the field just below the ionosphere is conveyed to the Earth by the random screen technique. This hybrid method was further extended by Maurits et al.  and Zernov et al.  to the conditions of the statistically nonstationary ionosphere, which are the most typical for the high-latitude and equatorial ionosphere containing mesoscale moving inhomogeneities (e.g., blobs, arcs, bubbles).
 To compensate for the background ionosphere in range-finding using satellite navigation systems, such as GPS, a linear combination of signals are used that travel the same path at the same time, but at differing frequencies, such as the L1 and L2 signals broadcast by GPS satellites. Previously, in the framework of the hybrid method, we addressed the assessment of the contribution of diffraction for range errors in the dual-frequency mode of operation and reported first results by Gherm et al. [2006b]. Here we consider this problem in more detail. As mentioned above, this treatment will not be confined to just the case of the weak scintillation of the field but will also account for strong scintillation by employing the hybrid method for the analysis. In particular, when extending the model, the effects of the mutual correlation of the fields at different frequencies will be taken into account in the procedures for generating the appropriate random time series. General relations and concepts will be presented in section 2 whereas, in the following sections, results and their analysis will be given.
2. Scintillation Propagation Model Extension
 To account for the effect of the field correlation at two different frequencies in the conditions of strong scintillation, the propagation model and simulator described by Gherm et al.  has been substantially extended. The version of the scintillation propagation model and simulator based on this model [Gherm et al., 2005] contains a physical model which takes as input models of both the background and stochastic (time-varying irregularities) ionosphere components. The time varying electron density fluctuations are specified in terms of their magnitude, velocity, outer scale (independently in three dimensions), aspect ratio and spectra index p which gives the inverse power law of their anisotropic spatial spectrum. The propagation model can treat scintillations for the case of very strong fluctuations of the electron density at GPS frequencies. As mentioned above, this is by means of a hybrid model which is a combination of the complex phase method together with an appropriately placed random screen below the ionosphere.
 To produce the two-frequency statistical moments of the random field, which are needed to treat the problem of the signal correlation at spaced frequencies and assess the diffraction errors in dual-frequency range finding for the case of strong scintillation, the scintillation propagation model and the simulator program have needed to be further modified. In the new version, random fields for each frequency are generated taking account of cross correlation of the phases and log-amplitudes for both frequencies. This stems from the fact that the fields of different frequencies traversing the same trajectories and, consequently, the same (random) inhomogeneities are not statistically independent, but, on the contrary, possess quite a strong correlation [Gherm et al., 2006b]. To implement this technique of generation of random realizations of the fields at different frequencies, the single-frequency statistical moments from Gherm et al.  and Maurits et al. 
have been supplemented by the appropriate two-frequency statistical moments of the complex phases of both the fields derived in the scope of the complex phase method for the fields on the screen, appropriately chosen just below the ionosphere:
Where ηn, ητ are the local spectral variables depending on the variable s along the reference ray which are expressed through spectral variables κn, κτ on the screen at s = s0 by means of the following set of linear relationships
Functions Δn(s) and Δτ(s) are written in ray-centered coordinate system components of the vector of the distance at the current point s between the rays arriving at s0 at the two different points spaced by the vector with the components Δn(s0) and Δτ(s0). J(s) is the Jacobian of transformation from transversal spectral variables at the point of observation to the transversal spectral variables in the local ray-centered coordinate system. The vacuum wave numbers are k1,2 = ω12/c. dn(s) and dτ(s) are the components of vectorial distance between the two rays corresponding to the different frequencies; if k1 = k2 then dn(s) = dτ(s) = 0. The function Bɛ(s;0,κn,κτ,k1,k2) is the three-dimensional spatial spectrum of the dielectric permittivity fluctuations with a zero value of the spectral variable Fourier-conjugated to the variable s along the path; it also depends on the field frequencies. The coefficients Dn, Dτ, and Dnτ are the elements of the matrix = (+)−1, i.e., the inverse to the matrix +, introduced by Gherm et al. . The matrix + is actually a sum of curvature matrices of the incident field and of the Green's function. These are obtained by integrating the corresponding differential equations along the reference ray and also depend on the variable s.
 Having calculated the functions above, the spectra of the two-frequency cross-correlation functions of the phase and log-amplitudes of the fields can be easily derived in the form
Functions Si and χi are the phase and log-amplitude of the field of the frequency ωi. Angle brackets denote the ensemble averaging, so that the expressions on the left-hand side of the equations (7) are the correlators required.
 In the numerical calculations, a turbulence model of the ionospheric fluctuations is considered with an anisotropic inverse power law spatial spectrum of the form:
Here CN2 is normalization coefficient, ɛ0(s,k) is 3-D distribution of the dielectric permittivity of the background ionosphere along the reference ray depending on frequency, σ2N(s) is the variance of the fractional electron density fluctuations, Ktg = 2πltg−1 and Ktr = 2πltr−1 where ltg and ltr are the outer scales of the turbulence along and perpendicular to the geomagnetic field direction, respectively. As is seen, two cross-magnetic field scales are introduced in (8), so that local random ionospheric inhomogeneities are presented as full 3-D bodies, and the direction of the cross field anisotropy is defined by the direction of ionospheric drift [Tereshchenko et al., 2005].
 The spectra of the two-frequency correlation functions (4) and (5) together with the single frequency ones (1)–(3) permit the proper generation of the two random two-dimensional correlated distributions of complex phases ψ1 = iS1 + χ1 and ψ2 = iS2 + χ2 below the ionosphere, and, consequently, the introduction of the random screens having random field distributions (or their random spatial spectra) just below the ionosphere.
 To briefly describe the procedure of generating the random screens, introduce, for convenience, the four-component vector x = (xi) = (S1, χ1, S2, χ2) comprising the desired random realizations of the spatial spectra of the field phases and log-amplitudes for both frequencies. Our purpose is to generate these realizations satisfying the correlation relations (1)–(7). To produce the random realizations possessing the correlations required, we introduce another four-component vector ξ = (ξ1, ξ2, ξ3, ξ4) with the components which are the realizations of random noncorrelated Gaussian fields with zero mean and unit variance, e.g., 〈ξjξl*〉 = δjl. Then the components of vector x are expressed through the components of vector ξ via the linear relation xi = aijξj, or, in matrix notation, x = Aξ where A is a matrix with elements aij.
 To obtain the elements aij we construct the covariance matrix B with elements bik
where the prime sign denotes a complex conjugate transpose operation.
 The elements bik are just the respective correlators (1)–(3), (7), so that the matrix B is real-valued and symmetric. As is known [see, e.g., Devroye, 1986, chapter 11], the covariance matrices are always symmetric positive-definite, and it is always possible to find a nonsingular matrix A such that AA′ = B. This fact justifies the procedure. Even more so, it is possible to decompose a symmetric positive definite matrix into a product of two triangular matrices, which are more useful in further calculations. Then, having obtained the matrix A, generating the random screens is straightforward employing the relation x = Aξ.
 In turn, the random spatial transforms of the fields are then conveyed down to the Earth's surface by the propagator
By this means, properly correlated random realizations of the field on the Earth's surface are obtained at two frequencies including the regime of strong scintillation. Then any required correlations and cross correlations of these fields and of their phases and amplitudes, in particular, can be calculated. This enables, for example, the extraction of the phases of both the fields to use them to analyze the range errors in the dual-frequency mode of operation due to fluctuations and diffraction effects on random inhomogeneities, in particular.
3. Analysis of the Ionosphere Free Range Errors
 Following Gherm et al. [2006a, 2006b], we write the phase path L really measured along the path of propagation of the GPS link as the following sum
Here ϕ is the measured phase and k is the wave number for vacuum. Quantity ρ is the true geometrical range and Δ represents the effects due to the ionosphere. Traditionally, quantity Δ is written in the geometrical optics approximation as follows:
Here n is the refractivity index of the ionosphere, which includes both the background and the fluctuational parts of the electron density of the ionosphere, ds denotes integration along a curved path of propagation, whereas ds0 stands for integration along the straight line of sight. The first item in the sum (12) gives the ionospheric effects integrated along the straight line of sight, and the second item in the square brackets represents the refraction error. The term in the square brackets is normally neglected as it introduces errors of the order of only millimeters, whereas the effects due to the first item are mainly excluded by means of the scheme of the dual-frequency operation (but phase advances/group delays due to higher-order terms in the refractive index binomial expansion are not eliminated). However, there always occurs the contribution of diffraction into the scattering of the field by random inhomogeneities of the ionosphere. Then to properly account for this diffraction, the propagation problem should be considered in terms of the full wave type of solution rather than in the geometrical optics approximation. Therefore, in contrast with the equation (12), quantities ϕ and Δ in equation (11) should be derived from the full wave solution to the problem of propagation in the fluctuating ionosphere
Then, performing the identical transformation as follows
shows that the part of phase, resulting from diffraction by irregularities is given by the quantity
The latter represents a pure diffraction effect, as δ = 0 in the geometrical optics approximation. In the dual-frequency method, an additional random range error due to the term δ from equation (15) is given by the equation
This quantity can be statistically assessed in terms of its variance (or r.m.s., which is the square root of the variance). This will be done in the next sections.
4. Results and Discussion
 In our technique we produce Δδ by means of a hybrid method [Gherm et al., 2005; Zernov et al., 2009] extended as described in section 2, which gives the random realizations of the field on the Earth's surface for two different frequencies, including the case of strong scintillation. Obtaining these then permits the further analysis. In particular, phases and amplitude of both the fields are extracted and processed in order to assess the diffraction errors and the mutual correlation of the phases at different frequencies.
 When discussing the correlation properties of the field phases at different frequencies, attention should be paid to the procedures of processing the experimental/simulated data in order to properly retrieve the random component of the field's phase. Fast phase changes accompanying deep fades have been found to be a particular problem [Gherm et al., 2003; Humphreys et al., 2010a, 2010b] and result in cycle slips or even loss of phase lock in the PLL. The occurrence of these cycle slips is readily understandable when considering the random phasor walks near to the origin in the complex plane, particularly for the conditions of strong scintillation.
 The random phasor R(ω, r) of the field is introduced as follows:
Here E0(ω, r) is the field in the absence of fluctuations of the ionospheric electron density, and χ(ω, r) and S(ω, r) are the log-amplitude and phase of a random field at frequency ω generated on the Earth's surface at a given location r according the technique described above in section 2. The time t appears instead of r in the approximation of the frozen drift of the random ionospheric inhomogeneities. The phasor is identically equal to unity if the fluctuations of the electron density in the ionosphere are not taken into account.
 To properly retrieve the phase of the randomly generated complex field, certain procedures have been introduced in the present extended version of the scintillation propagation model. These provide the continuity of the phase retrieved and allow for fast phase changes not exceeding half a cycle at a single sampling step. Several cycles between two frequencies may occur if a phasor tracks around the origin of the phasor complex plane (clockwise/or anticlockwise) and its trajectory has a loop containing the origin of the coordinate system. In these cases the phase difference between the phases of the two frequencies may accumulate an integer number of cycles (termed also as cycle slips, positive, or negative, depending on the direction of the rotation around the origin). All this will be illustrated and discussed in more detail in section 4.1.
4.1. Diffraction Range Errors
 To estimate the diffraction errors in dual-frequency range finding caused by scintillation it is vitally important to clearly understand the peculiar properties of the behavior of the random phasor (17) in the vicinity of the origin in the complex plane. To illustrate the details, the appropriate time series for different pairs of frequencies have been generated. In the calculations a fairly high electron density slant background profile has been chosen. In this model case, the profile of the background ionosphere was generated by the NeQuick ionospheric model, and corresponds to a slant TEC of 90 TEC units. The raypath was oriented along the magnetic meridian to the South looking up at the satellite from the ground at an elevation angle of 60°. Random inhomogeneities of the ionosphere were specified by the spatial spectrum of inverse power law of the form (8). The parameters of the spectrum were: spectral index p = 3.7, smallest outer scale 5 km, cross-magnetic field aspect ratio 5 and longitudinal aspect ratio 20. Calculations were performed for different pairs from the set of four frequencies (in MHz): 1575.42 (L1), 1227.60 (L2), 1381.05 (L3) and 1176.45 (L5). To provide different values of the scintillation index S4, only the variance of the fractional electron density of the ionosphere was changed. Where appropriate, the results of calculation will be presented as functions of S4 (determined for the frequency L2.)
 In Figures 1–4, results of calculations are presented obtained for the frequencies L1 (blue line) and L2 (red line). In Figure 1, the random time series for phase (Figure 1, top) and log-amplitude (Figure 1, bottom) are given. In Figure 1 (bottom), moments of time when deep fading occurs can be clearly seen. However, as can be seen in Figure 1 (top), a single cycle phase slip occurs only once at approximately 122 s. To distinguish this particular case from other cases of deep fading in the amplitude when no cycle slip occurs, the behavior of the random phasor (16) must be investigated in detail for these two different cases. These are presented in Figures 2 and 3.
Figure 2 corresponds to the portion of the 180 s record (Figure 1) for the 6 s interval (4–10 s), where deep fading occurs without a resultant single-cycle phase slip. Figure 3 shows the case of another 6 s interval (120–126 s) with a single-cycle slip of phase at the moment of deep fading at frequency L2. Figures 2 (right) and 3 (right) clearly show the different behavior of phases. Specifically, both phasors in Figure 2 (right) do not walk around the origin of the complex plane, whereas the phasor for the frequency L2 in Figure 3 (right) performs a full circle with the origin inside, which results in a single-cycle slip of the phase of the field at frequency L2 (corresponding to the accumulation of an extra 2π radians of the phase of the field at the L2 frequency).
 Different types of behavior of the random trajectories of the random phasor (17) can be solely attributed to the diffraction effects, and can only be properly modeled, when treating the propagation problem in the approximation of the full wave type solutions. They could never be observed, if instead, the GO approximation was utilized. Indeed, in the case of the GO approximation, it follows from (15) that
Here TEC(t) is the total electron content as a random function of time. From this it is clear that, in the GO approximation, there is no way that, for two different frequencies, the phase random walks can lie on different sides of the origin of its complex plane; this can only occur, as aforementioned, when the problem is considered in terms of the full wave solution taking account of the diffraction effects.
 According to the results presented in Figures 1–3, the random series for the diffraction range error from equation (15) is given in Figure 4. It is clear from Figure 4 that the results of calculating the statistical moments of the range error will substantially depend on the actual interval of averaging, and the statistics of the cycle slips occurrence must be studied. This requires the investigation of a large number of random time series generated according to the technique presented here. A similar statistics investigation was performed from an analysis of experimental data by Seo et al. . We will address this problem in more detail below but point out here that, when averaging over the time intervals not containing cycle slips in phase, we arrive at the quantities for the r.m.s. of the diffraction range error in the two-frequency measurements, reported by Gherm et al. [2006b]. These are of the order of up to about 25 mm for the L1/L2 pair in the conditions of strong scintillation (S4(L2) = 1.0).
 To obtain reliable values of the r.m.s. of the diffraction range error in the two-frequency measurements for the general case when phase slips can occur, a special procedure has been devised. A code was developed, which generates a hundred random time series for the quantity Δδ from equation (16) according to the procedure described above. When generated, every series of the random error Δδ has the duration of 180 s and may have a random number of cycle slips, each of 2πM radians (where M is a positive, or negative integer, or zero). It is worth pointing out here that in contrast to investigations where some empirical assumption respecting the probability density distribution of the field intensity scintillation is made [e.g., Humphreys et al., 2010a, 2010b] where a Nakagami distribution is utilized), we are not confined by an assumption of any particular distribution but produce the field realization on the basis of reasonably rigorous solutions to the appropriate stochastic equations governing the propagation processes in the fluctuating ionosphere as developed in our propagation scintillation model [Gherm et al., 2000, 2005; Maurits et al., 2008; Zernov et al., 2009].
 By averaging over the ensemble of 100 random series at each value of S4 for the L2 frequency, plots have been obtained and are presented in Figure 5, which show the dependence of the diffraction range error on the scintillation index S4 for the pairs of frequencies L1/L2, L1/L3, L1/L5, L2/L3 and L2/L5. As can be seen from Figure 5, the curves for different pairs of frequencies have common features, namely they all have a linear part when the fluctuations are weak (S4(L2) < 0.65), and diverge to larger values from the linear dependence as the fluctuations become stronger. The curves for pairs L1/L2 and L1/L5 practically coincide, while the curves for the pairs L1/L3, L2/L3, and, especially, L2/L5 show considerably larger errors. This is because, for the first two frequency pairs, the frequency separation is larger (348 and 399 kHz) resulting in a smaller error for the dual-frequency technique than for the other L1/L3 and L2/L3 (194 and 153 kHz) and L2/L5 (51 kHz) cases. In particular, as is seen from Figure 5, for the pair of frequencies L1/L2, the diffraction range error in the dual-frequency mode of operation may reaches 10–12 cm as the L2 frequency scintillation index approaches 0.9 for the ionospheric conditions and geometry given above.
 As our analysis shows, when the regime of weak scintillation (S4 up to 0.6–0.7) occurs and the range error shows a linear dependence on S4 (L2), the generated random time series never show cycle slips at any frequencies. The cycle slips occur in the case of stronger scintillations (S4 > 0.7), and this results in a strong nonlinear dependence of the diffraction range error on the severity of the scintillation. If, however, phase slips were removed in all the realizations in the ensemble of 100 samples for averaging (or if they do not occur), all the curves in Figure 5 would become identical to those given by Gherm et al. [2006b, Figure 4].
 To conclude this section, it should be mentioned that the procedure of averaging essentially defines the value of the predicted diffractional error in the dual-frequency range measurements. When, as was done here, the cycle slips were taken into account the predicted value of the error may reach up to 200 cm for L2/L5 and 20 cm for L1/L2 for the ionospheric conditions and the geometry of propagation considered.
4.2. Cross Correlation of Phases at Different Frequencies
 To quantitatively characterize the correlation of signals for different frequencies, the correlation coefficient of phase is defined by the equation
where ϕ1 and ϕ2 are the phases for frequencies 1 and 2, respectively. In the GO approximation this quantity is identically equal to unity: it only becomes smaller when the diffraction effects are taken into account in the appropriate calculations.
 As mentioned in section 4.1, the results of calculation of the statistical moments, and of the cross-correlation coefficients, in particular, will strongly depend on the actual interval of averaging, and the statistics of the cycle slips, or accumulated cycles. Again, if the calculation of the moment (19) is performed over the random phase time series with no cycle slips, the curves for the correlation coefficient dependence on S4 for the different frequency pairs can be obtained as given by Gherm et al. [2006b] and are characterized by a very high level of correlation.
 To obtain here the more general results we again used a code enabling averaging over the ensemble of a hundred pairs of time series generated for different frequency pairs, where the different realizations have differing amounts of the accumulated phase cycles. First, the two-frequency mutual correlation functions was calculated as the time convolution integral for each of the hundred pairs of the ensemble, which were normalized to give the value of the appropriate cross-correlation coefficient at t = 0. These functions are plotted in Figure 6 for the conditions of propagation corresponding to S4 (L1) = 0.62; (S4(L2) = 0.89). Figure 6 (left) shows the phase cross-correlation functions, and Figure 6 (right) shows the amplitude cross-correlation functions. The highest curve of the hundred in Figure 6 (left) corresponds to the case of no cycle slips in the time series for both frequencies and has a very high value of the cross-correlation coefficient as was given by Gherm et al. [2006b]. The lowest curve gives the lowest level of the cross-correlation coefficient of the order 0.3, which is an extreme random result, obtained for this particular realization. It is close to the correlation coefficient obtained by Béniguel and Adam  for similar values of the scintillation indexes for the pair of frequencies L1/L5. The value of 0.3 is the lowest one for our ensemble of a hundred pairs of realizations, whereas averaging over all the curves gives the “mean” cross-correlation function (red curve in Figure 6, left) with its maximum value of 0.88, which can be considered to be the “mean” cross-correlation coefficient. This is the average value of the cross-correlation coefficient for the frequency pair L1/L2 obtained for the scintillation indexes given above in this paragraph. For this case the characteristic cross-correlation time of the “mean” phase cross-correlation function is about 12–13 s.
Figure 6 (right) shows the time cross-correlation functions of the amplitudes of the fields at two frequencies L1 and L2. These functions do not vary essentially from one realization to another. The cross-correlation coefficients for the amplitudes are given by the values of these functions at the moment of time t = 0. The time decorrelation of the field amplitudes can be seen to be much smaller than the phase decorrelation time, and the “mean” characteristic timescale is of the order of 1–2 s, shown by the red curve.
 Finally, an investigation was made of how the phase cross-correlation coefficients depend on the severity of scintillation. The results of the calculations are presented in Figure 7 where the correlation coefficients for different pairs of frequencies are plotted against the scintillation index S4 for the L2 frequency. It should be mentioned again that, as is seen from the graphs in Figure 7, in the conditions of weak scintillation (no cycle slip case when S4 < 0.6–0.7) the “mean” cross-correlation coefficients differ only slightly from unity. When the values of S4 become large and the intervals of averaging contain cycle slips, this results in a substantial reduction in the “mean” values of the coefficients of cross correlation down to values of around 0.8, the smallest value of 0.79 being for the pair of frequencies L2/L5. When, in the averaging procedure, the cycle slips were removed in all the ensemble realizations, the plots in Figure 7 become the same as presented by Gherm et al. [2006b] (when making this comparison, note the different scale).
 Finally, it should be pointed out that the central remit of our paper is the propagation effects so that we are, in effect, determining fields at the input to the GNSS receiver. We do not therefore consider any effects taking place inside the receiver where, for example, fast phase changes and deep amplitude fades can also lead to cycle slips and phase loss of lock. Such hardware effects are, however, linked to the results of this paper in as much as the propagation mechanisms elucidated herein can well be the cause of the correlated fast phase changes and deep amplitude fades which cause phase lock loss [see Gherm et al., 2003]. A treatment also incorporating such effects is much more complex as it must, of necessity, include also the hardware operation of the receiver (in particular the phase tracking operation) and, for dual-frequency GNSS, the respective CNRs of the different frequency signals may also be a significant factor.
 A number of the effects have been investigated which result exclusively from diffraction on local random inhomogeneities of the ionosphere. To treat these, in particular, in the case of strong scintillation, the simulator program based on the hybrid method has been further modified to simulate the fields for both frequencies on the ground, taking account of their cross correlation. Employing this modified technique, the errors in the two-frequency range finding method caused by ionospheric scintillation have been estimated for particular ionospheric conditions and a realistic fully three-dimensional model of the ionospheric turbulence and the cross-correlation properties of the phases at different frequencies have been studied. The results are plotted against the strength of the electron density fluctuations for five different pairs of frequencies (L1/L2, L1/L3, L1/L5, L2/L3, and L2/L5). The results show that the dependence of the diffractional error on the scintillation index diverges from a linear relationship, the stronger are scintillation effects, and may reach up to some tens of centimeters. It has also been shown that the values of the correlation coefficients for different pairs of frequencies are fairly close to unity in the conditions of weak scintillation but reduce substantially as the variance of the electron density fluctuations increases. The procedure of averaging the modeled/experimental phase records substantially defines the reliability of prediction of the diffractional range error and the cross-correlation coefficients. This needs to be taken into account for dual-frequency ionospheric correction or other related positioning methods in scintillation conditions. The extended hybrid technique, described and utilized above, allows for the assessment of the ionosphere-free range diffraction errors and the different frequency phase cross correlations even in the conditions of a severely disturbed ionosphere which can be caused by the presence of mesoscale ionospheric inhomogeneities as in the case of the equatorial bubbles, or similar scale inhomogeneities in the high-latitude ionosphere.
 The authors would like to acknowledge some financial support for this research from the UK Engineering and Physical Sciences Research Council (EPSRC) through grant EP/H004637/1.