^{1, 2}

^{1}

^{2}

We have used numerical simulation to study the effects of ionospheric irregularities on accuracy of global navigation satellite system (GNSS) measurements, using ionosphere-free (in atmospheric research) and geometry-free (in ionospheric research) dual-frequency phase combinations. It is known that elimination of these effects from multifrequency GNSS measurements is handi-capped by diffraction effects during signal propagation through turbulent ionospheric plasma with the inner scale being smaller than the Fresnel radius. We demonstrated the possibility of reducing the residual ionospheric error in dual-frequency GNSS remote sensing in ionosphere-free combination by Fresnel inversion. The inversion parameter, the distance to the virtual screen, may be selected from the minimum of amplitude fluctuations. This suggests the possibility of improving the accuracy of GNSS remote sensing in meteorology. In the study of ionospheric disturbances with the aid of geometry-free combination, the Fresnel inversion eliminates only the third-order error. To eliminate the random TEC component which, like the measured average TEC, is the first-order correction, we should use temporal filtering (averaging).

At present, remote sensing methods relying on ground-based and low-orbit observations of signals from global navigation satellite systems (GNSS) such as GLONASS and GPS are finding ever-widening application in environmental research [

GNSS remote sensing methods (see, e.g., [

In order to find the pseudo distance

The dual-frequency measurements from formula (

The second-order correction which takes into account geomagnetic effects on the ionospheric refractive index is ignored in dual-frequency measurements (

After accounting for the second-order residual error, further accuracy improvement is associated with the consideration of the third-order error. In the geometrical optics approximation, this error is largely attributed to the path deviation from the straight line and can be eliminated in triple-frequency measurements [

The above results of studies of GNSS ionospheric errors largely refer to the results obtained with ionosphere-free combination (

Consider the behavior of GNSS signal in the form of harmonic wave with the time dependence

Thus, the errors to be considered are associated in the geometrical optics approximation with the GNSS ray bending. Moreover, it is necessary to correctly describe the interaction of the wave field with the irregularities for which conditions of the geometrical optics approximation (

The electron density of ionospheric plasma is normally represented as sum of two components

The electron density

Random ionospheric irregularities are usually represented by a random, quasi-homogeneous, normally distributed field with a given spectrum

Spectrum (

By substituting (

Note that contrary to the second term, the first term on the right-hand side in (

After substituting (

By substituting (

Note that in (

Now it is easy to find statistical characteristics of dual-frequency measurement errors. For measurements with ionosphere-free combination (

For measurements with geometry-free combination (

To assess probable errors in dual-frequency measurements, we have performed numerical simulation with (^{2}). Random ionospheric irregularities were specified by spectrum (

Figures

Elevation angle dependence of average (a) and standard deviations (b) of corrections of first (dashed line) and third (solid line) orders for ionosphere-free combination (

The same as in Figure

A characteristic property of absolute average corrections and standard deviations is their increase with decreasing elevation angle (see Figures

In [

As is well known, with the Green function at a given wave field on the plane where the wave leaves the inhomogeneous medium, it is possible to find the field on another, distant plane. When the medium between these planes is homogeneous, we can take the field of point source in free space as the Green function. This Kirchhoff approximation is widely used for analyzing wave propagation behind the slightly inhomogeneous layer. Yet the wave field where the wave leaves this layer can be described using the geometrical optics approximation. The Kirchhoff formula allows us to calculate this field in the distant region exhibiting various diffraction effects, caustics, and strong fluctuations for which the geometrical optics approximation does not work. Since the small-angle approximation is applicable to large-scale irregularities, the Kirchhoff formula takes the form of Fresnel transform [

When the inhomogeneous layer is sufficiently thin, it modulates only the transmitted wave phase

On the other hand, inversion of Fresnel transform (

Substituting second Rytov approximation (

Figure

Virtual screen position dependencies of the average (a) and standard deviation (b) of corrections of first (dashed line) and third (solid line) orders for ionosphere-free combination (

Figure

Virtual screen position dependencies of the scintillation index for the frequency L1 (

In Figure

The same as in Figure

In this study, ionospheric errors of GNSS remote sensing using both ionosphere-free and geometry-free combinations were analyzed. In the former case, we demonstrated the possibility of reducing residual ionospheric errors in ionosphere-free combination by Fresnel inversion. The inversion parameter, the distance to the virtual screen, may be selected from the minimum of amplitude fluctuations. This suggests the possibility of improving the accuracy of GNSS remote sensing in meteorology.

In investigations into ionospheric disturbances through geometry-free combinations, the Fresnel inversion eliminates only the third-order correction. To eliminate the random TEC component which, like the measured average TEC, is the first-order correction, we should use temporal filtering (averaging). However, measurements of this random TEC component allow us to examine the fine structure of ionospheric plasma via GNSS.

The Fresnel spatial processing requires constructing a phase GNSS array of sufficiently large sizes. Despite the difficulty in constructing this array, it seems necessary to develop the array, taking into account feasibility of using it in many fields of research on the troposphere, ionosphere, earthquakes, and volcanic activity.

The author declares that there is no conflict of interests regarding the publication of this paper.

The author is very grateful to O. A. Kulish for her assistance in preparing the English version of the paper. This study was supported by the grant from the Russian Scientific Foundation (Project no. 14-37-00027).