Methods for describing the field of ionospheric waves and

spatial signal processing in the diagnosis of inhomogeneous

ionosphereMikhail V. Tinin

Irkutsk State University, 20 Gagarin blvd, Irkutsk, 664003, Russia, e-mail:

mtinin@api.isu.ru

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.

Perturbation theory in the wave problem.

20 0 0( , ) ( ) ( , ) ( )U k U f r r r r r r r

( ) ( ) ( ) r r r ( ) ( ) r r

0 0 0 1 0 2 0 3 0

2 30 0 0 0

4 3 30 0

60

( , ) ( , ) ( , ) ( , ) ( , ) ...

( , ) ( ') ( , ') ( ', ) '

( , ') ( ', '') ( ') ( '') ( '', ) ' ''

( , ') ( ', '') ( '', ''') ( ') ( '') ( ''') (

V

V V

U U U U U

U k G U d r

k G G U d r d r

k G G G U

r r r r r r r r r r

r r r r r r r

r r r r r r r r

r r r r r r r r r r

3 3 30''', ) ' '' ''' ...

V V V

d r d r d r r

0r

'r

''r 'r

0 0( , )U r r2 0( , )U r r

1 0( , )U r r

Born approximation in the wave problem.

0 0 0 1 0

20 3

0 0

( , ) ( , ) ( , )

exp exp ( ')' '

4 ' 'V

U U U

ik ikk AA d r

r r r r r r

r r rr

r r r r r r

0( ') ' ' r r r r r

'r r

'r

0r 0'r r

in

sn

O

2 2

1 00

2 exp( , )

k A ikU

rr

r r Q

0r r s ik Q n n

Geometrical optics approximation.

( )

0

( )exp exp

( )

n

n

AU A ik ik

ik

rr r r r

2 r(0) (0)2 ( ) ( ) 0A A r r

(0)A A

1, ( )

2

d d

d d

r pp r

0

0

( ) ( ') 'd

r r

( )d

d

r

00( ( )) ( (0))

daA A

da

rr r

r

( ) ( ) ( ) 1 ( ) r r r r

00

1 0

exp ( , )1 ,

UU ik

D

rr ρ ρ

ρ ρ

0

0 0 0( , ) 0.5 ' / ' / , ' 'tz

tzz z Z z z Z z dz ρ ρ ρ ρ

2

1 0 00

, , , , , ( , )x y z z D Z

r ρ ρ ρ ρ ρ

ρ ρ

0, ( ')( ' ) /Fl l a z z z z Z

0tZ z z

First approximation.

First Rytov approximation

0 expU U r r r

0' ( ')( ' ) /z z z z z Z

0

2220

0

' ', '' exp ' ' '

4 ' 2 '

z

t

z

d zk ikdz z z z z

z z Z Z

ρ ρρρ

Imk 0 exp ReA A

0( ')( ' ) /Fl a z z z z Z Rytov approx. GO

Phase screen approximation

2

'

exp1, ', ' '

2 'S

ikRU z U z d

z R

ρ ρ

2 2

2

' '

'1'

2 '

R z z

z zz z

ρ ρ

ρ ρ

exp

p p

p p p p

U V dp

A ik dp

r r

r r

Integral representations

2

2

'

exp ' '1, exp ', ' exp '

2 ' 2 'S

ik ik z zU z A ik z ik d

z z z z

ρ ρ

ρ ρ

', ' exp ', 'U z A ik zρ ρ

1kl

Small-angle approximation

0 0 0

0

( , ) 0,5 ' / ' / , ' 't

t

zz z Z z z Z z dz

z ξ ξ ξ ξ

0( ')( ' ) /Fl a z z z z Z DWFT GO

20 0 0( , ) ( ) ( , ) ( )U k U r r r r r r r

0 0( , ) ( , ) exp ,sU U ikZr r r r 2

22

2 ( ) 1 0s ss

U Uik k U

z

r

The double-weighted Fourier transform (DWFT)

220

0 03 3

2 20 0 0 0 0

( , ) exp ( ) / 24

exp [2( - - ) / ( , )] ,

ZA k

U ik ZZ

d d ik Z

ρ ρ ρ ρ

ξξ ξρ ξ ρ ξ ξ

1kl

Spatial processing by DWFP

22 2 0

0 0 0 0

02

.

( )2ˆ ( , ) ( , ) exp ( * * )2

2 exp ( )

4

ikU d d U ik ikZ

Z Z

A Zik

k Z

0

00

ρ ρρ* ρ * ρ ρ ρ ρ ρ ρ

ρ*ρ *ρ*,ρ *

0

0, 0.1, 1 , 3 ,

3 , 2 , 5.4c c m

F

z l см z м

z м мм a см

ρ

• 2) analyzing vertical sounding of the ionosphere.

Now consider the applications of the above methods for:

• 1) reducing errors of GNSS measurements;

21 2 3 41 2 3

1

/ 40,3 / 40,3 / 40,3 / +...igo i g g g

i

D d f D I f I f I f

Second-order correction

0

1 ( ') 'z

go

z

I N z dz0

2 ( ') ( ') cos ( ') 'z

go H

z

I N z f z z dz

1

0 0

2

3 1

0

, '1 '

2

zz

go

z z

N zI dz dz

The Ionospheric Error of the Navigation System within the Geometrical Optics Second Approximation

First-order correctionThird-order correction

The first approximation: the dual-frequency

measurements

22 3 41 2 3

21

40,3 / 40,3 / 40,3 /

40.3 /

go g g g

g

D I f I f I f

D I f

2 2(2) 1 1 2 2

2 21 2

f f

Df f

For the dual-frequency measurements ionosphere-free combination eliminates the first-order ionospheric error- most of the ionospheric error.

1 1 2 2( ), ( )go gof f

Given only the first-order correction, we obtain first approximation

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

2 2(2) 1 2 1 2

2 21 240

3

.

f fI

f f

The Ionospheric Errors of the Dual-Frequency Navigation System within

the Geometrical Optics Approximation

(2) (1) , D D D 2 3 , go goD D D

2 2 1 2 1 240.3 / , go goD I f f f f 2

3 1 2 340.3 / .go goD f f I

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.

Modeling statistical characteristics of the residual error for a turbulent ionosphere

/m er h R H

2

( ) exp 0.5 1 exp 80.6

cfN r

2 3

/22 2 2 2003/2

( ) ( / 2)( ) exp /

(( 3) / 2)

pp

NN m

z p

p

κ

Chapman layer

0 0 0

0 0

2 / , 5.92 / ,

15 ., 320 ., H=70 ., 1 .,

20 ., 1 , 0, 11 / 3, 0.05.

m m N

c m H

m

L l N

f МHz h km km f MHz

L km l km p

r r

Is it possible to eliminate higher ionospheric errors

with the linear combination of measurements at more

frequencies?

Second-order correction

First-order correctionThird-order correction

22 3 41 2 340,3 / 40,3 / 40,3 / go g g gD I f I f I f

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):

22 3 41 2 3

2 3 2 31 2 1

21

40,3 / 40,3 / 40,3 /

40.3 40.3 40.3 ( )cos ( )

40.3

g g g g

g g g H m m

g e

f D I f I f I f

D I f I f D I f f f z z

D I f

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):

, ( ) cos ( ) / 2e H H H m mf f f f f z z

(2) 2 2 2 21 2 1 1 2 2 1 2( , ) /e e e eD D f f f f f f

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:

22 41 1 1 1 3 1

22 42 2 1 2 3 2

22 43 3 1 3 3 3

40.3 40.3 ,

40.3 40.3 ,

40.3 40.3 .

g g e g e

g g e g e

g g e g e

f D I f I f

f D I f I f

f D I f I f

(3) 2 2 21 2 3 1 1 1 2 2 2 3 3 3( , , ) e e eD D f f f a f a f a f

2 2 2 2 2 2 2 2 2 21 1 1 2 1 3 2 2 2 1 2 3

2 2 2 2 23 3 3 1 3 2

/ , / ,

/ .

e e e e e e e e e e

e e e e e

a f f f f f a f f f f f

a f f f f f

By solving the system, we get a triple-frequency distance formula

Diffraction effects in GNSS measurements

2

1 2 32 3 2

40.3 40.3 40.3 f D I f I f I f

f f f

0

22

1

'' , ' cos

2

z

z

z zI dz d N z

k

κ

0 0 0

' '2 2

3 1 2 1 2 1 2 1 1 2 2

2 21 2 1 2

1' , ,

2

cos , , ', , , , , 2 , exp

z z z

z z z

k

I dz dz dz d d N z N z

S z z z k N z d N z i

κ κ κ κ

κ κ κ ρ κρ

2 21 2 1 2 1 1 2 2 1 2, , ', , , 0.5 0.5 ' /S z z z k z z z z z z k κ κ κ κ

Second-order Rytov approximation

The same as above at inner scale is 70 m

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.

The influence of diffraction effects on the first-order (dashed lines) and the third-order (solid lines) corrections

a

a

b

b

Fresnel inversion

Scintillation index for L1 (solid line) and L2 (dashed line) GPS signals as a function of virtual screen position

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

2

2

exp',

2

'1, exp

2t

bb

b

Sb

ik ik z zU z

z z

U z ik dz z

ρ ρ

Wave reflection from a layer with random inhomogeneities

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

3/233/2 3 34

0

0

2

0 1

4 2,

exp / 2 / 2 2 ,

i

G e d d sd pAk k

ik

0

r,r s,p

r r sp sr pr s,p

0 0

0

/ 2 ,G i k U d

r,r r,r

1 1

0 0

, 1/ 2 ' ' ' ' ' 2 ' ' 'd d

s,p p s r r p p

1( ) 1 ( ) r r

22 ( ) 0U

ik U k U

r 00

U

r,r

, ,, 1

' ' 1' , ' ' ' '

' ' 4

d d

d d

r pp p s r r

r

9/239/2 3 34

0 0

0

2 2 30

0 0 1 0

4

2

2 ' 'exp , , ,

2 2 4 96

i kG e d d d

k

ik z z

0

r,r

r rξ r ξ r ξ ξ

1 0 1 0

0

, , 1/ 2 1 '/ ' ' ' '/ 4 'z d

ξ ξ ς ς e

1 1( ) 1 ( ) 1 ' z r r r

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

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.. - 2000. - V.35, №6. – P. 1315-1322.

• M.V. Tinin, Yu.A. Kravtsov. Super – Fresnel resolution of plasma in homogeneities by electromagnetic sounding. Plasma Phys. Control. Fusion. – 2008. - V. 50. - 035010 (12pp). - DOI: 10.1088/0741-3335/50/3/035010.

• M.V. Tinin, B.C.Kim Suppressing amplitude fluctuations of the wave propagating in a randomly inhomogeneous medium Waves in Random and Complex Media. – 2011. - 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. 391-398  

• 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. 189-199, 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 10.1007/s00190-010-0425-z.

Thank you!