## 1. Introduction

[2] 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 *L*_{2} 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.

[3] 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.

[4] 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* [2007], 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.

[5] In the work by *Gherm et al.* [2000] 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.

[6] 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.* [2008] and *Zernov et al.* [2009] 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).

[7] 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.