1. Methods for describing the field of ionospheric waves and spatial signal processing in the diagnosis of inhomogeneous ionosphere
Mikhail V. Tinin
Irkutsk State University, 20 Gagarin blvd, Irkutsk, , Russia,

2. We consider the possibilities of application of both classical and new methods for the description of wave propagation to solving some problems of ionospheric propagation of radio waves.

3. Perturbation theory in the wave problem.

4. Born approximation in the wave problem.

5. Geometrical optics approximation.

6. First approximation.

7. First Rytov approximationGO

8. Phase screen approximation
Integral representations
Phase screen approximation

9. Small-angle approximation
The double-weighted Fourier transform (DWFT)
DWFT GO

10. Spatial processing by DWFP

11. Now consider the applications of the above methods for:
1) reducing errors of GNSS measurements;
2) analyzing vertical sounding of the ionosphere.

12. Second-order correction
The Ionospheric Error of the Navigation System
within the Geometrical Optics Second Approximation
Third-order correction
First-order correction
Second-order correction
The phase path of ionospheric radio waves in the geometrical optics approximation is equal to . On the right side of (1), the first term is the distance between a satellite and an observer. The second term (the first-order correction) is proportional to the total electron content. It is related to the difference between the phase velocity in isotropic plasma and the speed of light in free space. The third term (the second-order correction) accounts for the geomagnetic field effect on this velocity. The fourth term (the third-order correction) is mainly associated with the ray bending in the inhomogeneous ionosphere.

13. The first approximation: the dual-frequency
measurements
Given only the first-order correction, we obtain first approximation
For the dual-frequency measurements ionosphere-free combination eliminates the first-order ionospheric error- most of the ionospheric error.
Given only the first-order correction, we obtain first approximation. For the dual-frequency measurements ionosphere-free combination (3) eliminates the first-order ionospheric error- most of the ionospheric error. However, in some high-precision measurements the accuracy of order of 10 cm is no longer adequate.
As we can see the residual error is determined by the geomagnetic field contribution, the ray bending in the inhomogeneous ionosphere, and the diffraction effects.
As we can see the residual error is determined by the geomagnetic field contribution, the ray bending in the inhomogeneous ionosphere, and the diffraction effects

14. The Ionospheric Errors of the Dual-Frequency Navigation System within
the Geometrical Optics Approximation
To obtain the residual error of dual-frequency reception within the Geometrical Optics approximation, we substitute expression for the phase path of ionospheric radio wave in ionosphere-free combination.
To obtain the residual error of dual-frequency reception within the Geometrical Optics approximation, we substitute expression for the phase path of ionospheric radio wave (1) in ionosphere-free combination (3).
Thus we get the residual error as sum of second - and third -order errors of dual-frequency measurements (3). 14

15. Modeling statistical characteristics of the residual error for a turbulent ionosphere
Chapman layer 15

16. Second-order correction
Is it possible to eliminate higher ionospheric errors with the linear combination of measurements at more frequencies?
First-order correction
Third-order correction
Second-order correction 16

17. Approximate formula for the second-order correction
Approximate formula for the second-order correction. Eliminating the second-order effects
The geomagnetic field changes slowly at ionospheric heights. The third term on the right side can therefore be written as (Bassiri and Hajj, 1992, 1993):
So, by changing coefficients of the ionosphere-free linear combination, we can eliminate both first- and second-order effects in dual-frequency measurements (for details see report of E.V. Konetskaya):

18. The triple-frequency GNSS measurements: Eliminating the third-order effects.
Given phase measurements at three frequencies accounting for second-order effects, as above, we can write a system of equations:
But the ionospheric inhomogeneities with sizes less than the Fresnel scale produce diffraction effects.
By solving the system, we get a triple-frequency distance
formula

19. Diffraction effects in GNSS measurements
Second-order Rytov
approximation
To obtain the diffraction variant of formulas for the higher-order errors, we applied second-order Rytov approximation for the phase path.
As we can see, the consideration of the diffraction effects gives that the phase path is not of the form of a series in inverse power of frequency.
When the minimum size of inhomogeneities exceeds the Fresnel scale diffraction formulas are transformed into corresponding geometrical optics expressions.
To calculate the residual error of dual-frequency measurements, accounting for the diffraction errors, we substitute Eq. (9) in ionosphere-free combination 19

20. The same as above at inner scale is 70 m
The influence of diffraction effects on the first-order (dashed lines) and the third-order (solid lines) corrections a b
The angle-of-elevation dependencies of the average third-order correction (a) and of the standard deviations (b) of the corrections at inner scale is 1 km . Green and red lines correspond to the dual-frequency and tripe-frequency GNSS measurements respectively. b a
The same as above at inner scale is 70 m 20

21. Fresnel inversion Bias and standard deviation of
residual error of first (dashed line) and third (solid line) orders with two-frequency (green lines) and three-frequency (red lines) cases as a function of virtual screen position
Scintillation index for L1 (solid line) and L2 (dashed line) GPS signals as a function of virtual screen position

22. Wave reflection from a layer with random inhomogeneities
П.В. Силин, А.В Зализовский, Ю.M. Ямпольский Эффекты F-рассеяния на антарктической станции «Академик Вернадский». Paдиофизика и радиоастрономия 2005, T. 10, N1, C. 30­37

23. DWFT beyond the small-angle approximation; the method of Fock proper time

25. Conclusions
Increasing the accuracy of the known methods associated with the higher-order approximations, allows us to estimate the accuracy of GNSS measurements and suggest ways to improve it.
The development of new technical possibilities of the ionospheric plasma diagnostics requires a corresponding development of physically-based diagnostic methods
The methods considered for the description field of the probe signal can develop new ways to coherent quasi-optimal space-time processing, and find their application in the diagnosis of the ionospheric plasma and plasma fusion

26. See also
Yu.A. Kravtsov, M.V. Tinin. Representation of a wave field in a randomly inhomogeneous medium in the form of the double-weighted Fourier transform. Radio Sci V.35, №6. – P
M.V. Tinin, Yu.A. Kravtsov. Super – Fresnel resolution of plasma in homogeneities by electromagnetic sounding. Plasma Phys. Control. Fusion. – V (12pp). - DOI: / /50/3/
M.V. Tinin, B.C.Kim Suppressing amplitude fluctuations of the wave propagating in a randomly inhomogeneous medium Waves in Random and Complex Media. – V. 21, № 4. –P. 645–656.
M. V. Tinin, Integral representation of the field of the wave propagating in a medium with large-scale irregularities. Radiophysics and quantum electronics - 2012  V. 55   P  
Yu. A. Kravtsov, M. V. Tinin, and S. I. Knizhnin Diffraction Tomography of Inhomogeneous Medium in the Presence of Strong Phase Variations Journal of Communications Technology and Electronics, 2011, Vol. 56, No. 7, pp. 831–837
B. C. Kim and M. V. Tinin, “Contribution of ionospheric irregularities to the error of dual-frequency GNSS positioning,” J. Geod., vol. 81, pp , 2007.
B. C. Kim and M. V. Tinin, “The association of the residual error of dual-frequency Global Navigation Satellite Systems with ionospheric turbulence parameters,” JASTP, vol. 71, pp. 1967–1973, 2009.
B.C. Kim, and M.V. Tinin, “Potentialities of multifrequency ionospheric correction in Global Navigation Satellite Systems,” J Geod., 85, 2011, pp. DOI /s z.
Integral representation of the field of the wave propagating in a medium with large-scale irregularities. Radiophysics and quantum electronics   Volume: 55   Issue: 6   Pages:   Published: NOV 2012

27. Thank you!