18/04/2318/04/23

SOCIETÁ ITALIANA DI FISICA XCVII CONGRESSO NAZIONALE

L’AQUILA, 26-30 SETTEMBRE, 2011

A. Settimi *, C. Bianchi, C. Scotto, A. Azzarone, A. Lozito

Istituto Nazionale di Geofisica e Vulcanologia, Sezione di Geomagnetismo, Aeronomia e Geofisica Ambientale,via di Vigna Murata 605, 00143 Rome, Italy

* e-mail: alessandro.settimi@ingv.it

11

SEZIONE 4a : Geofisica e fisica dell'ambiente

A METHOD TO TEST HF RAY TRACING ALGORITHM IN THE IONOSPHERE BY MEANS OF THE VIRTUAL TIME DELAY

22

ABSTRACT

As well known,a 3D ray tracing algorithm furnishes the ray’s coordinates, the three components of the wave vector and the calculated group time delay of the wave along the path.

This last quantity can be compared with the measured group time delay to check the performance of the algorithm. Simulating a perfect reflector at an altitude equal to the virtual height of reflection,the virtual time delay is assumed as a measured time delay.

We find a very small percentage relative error between the calculated and the virtual delays, under the hypothesis of a monotonic electronic density profilefor both analytic and numerical models.

33

INDEX

1. Introduction.

2. 3D ray tracing algorithm: calculated group time delay.

3. Perfect reflector: virtual time delay.

4. Analytic and numerical electronic density profiles: percentage relative error of delay.

5. Appendix: checking virtual vertical (oblique) height.

6. Conclusions.

44

Introduction

In case of a real measurement, it is possible to compare the calculated group time delay tcalc with the measured time delay treal. When the differences are within an acceptable error due to discrete step of the mathematical process and the numerical ionospheric model,we assume that both 3D ray tracing algorithm and ionospheric model work properly.Nevertheless, the measured delay is usually not available for ionospheric 3D ray tracing users since real measurements of this quantity are quite uncommon. In order to test the algorithm performance,an alternative method to obtain something similar to the term treal is thus required.Propagation theorems and refraction laws, widely used in ionospheric physics applications, have been exploited at this aim.

In this talk,we deal about a 3D ray tracing algorithm derived from 6 differential equations with Hamiltonian formalism in geocentric spherical coordinates.The ray tracing program is written in MATLAB language for what concern input and output routine, while the executive computation nucleus derives from the FORTRAN program of Jones and Stephenson (1974).

55

3D ray tracing algorithm: calculated group time delay.

2

2 2 2 22

1( , , , , , , , ) Re

2r r

cH r t k k k k k k n

The independent variable τ is suitably chosen as a parameter satisfying the monotonocity condition along the ray path.

The Hamiltonian H is a propagation constant during the ray path.

ct

1

1

sin

r

dr H

d k

d H

d r k

d H

d r k

dt H

d

sin

1( cos )

1( sin cos )

sin

rdk H d dk k

d r d ddk H dr d

k k rd r d ddk H dr d

k k rd r d d

d H

d t

66c t calc i

i

t t

The 3D ray tracing algorithm ends the integration when the ray reaches the ground after reflection.

(1)

77

Perfect reflector: virtual time delay.

Under the assumption of a flat layered ionosphere:

we image to put a perfect reflector at an altitude equal to the vertical virtual height of reflection (i.e. compatible with a path in which the wave velocity of the light is c).

For a given high frequency (HF),this quantity can be simply calculated analytically or directly from the ionogram (Davies, 1990).

88

The measured group time delay treal can be compared to the virtual time delay tvirt.

This last quantity can be obtained immediatelyusing a pure geometrical relation if we consider a flat layering ionosphere or approximately spherical surfaces, i.e. electronic density profile dependent only on the quote, without horizontal gradient. For the validity of the relation treal= tvirt, a monotonic increasing trend of the electronic density profile is also required otherwise the valley between E and F layers invalidates the relation.

99

In an oblique path,from the transmitter positioned in the point indicated with A,through the ionosphere,the ray reaches the point B on the ground surface.

The ray tracing algorithm furnishes in output the arrival point B and the calculatedgroup time delay tcalc.

We simulate a perfect virtual reflector at an altitude h’v ,compatible with the virtual propagation, i.e. at the vertex C of a triangle ACB representing the oblique virtual path.

1010

Secant law

secob vf f fob is the oblique frequency,fv is the vertical frequencyand φ is the incidence angle.

Martyn’s theorem

The oblique virtual height hob of reflection at the oblique frequency fob

is equal to the vertical virtual height h’v of reflection at the vertical frequency fv , according to the secant law:

'( ) ( )ob ob v vh f h f

Breit&Tuve’s theorem

calc virtt t

The calculated group time delay tcalc of the wave propagating in the effective path at group velocity compatible with the refractive index is equal to the virtual time delay tvirt of the wave propagating along the oblique virtual path at the speed c of light in vacuum

1111

The virtual time delay tvirt can be immediately obtained,using a pure geometrical relation:

2virt

dt

c

(2)

1212

Analytic and numerical electronic density profiles: percentage relative error of delay.

1

2

3

4

5

6

7

8

9

10

11

12

3 6 9 12 15 18 21 24 27 300

0.5

1.0

1.5

2.0

2.5

3.0

BREIT&TUVE ( virtual tvirt

) - 18 degrees // - 30 degrees // - 45 degrees3D RAY-TRACING ( calculated t

calc ) - 18 degrees

// - 30 degrees // - 45 degreest

error ( % ) = t

calc - t

virt / t

virt - 18 degrees

// - 30 degrees // - 45 degrees

f ( MHz )

GR

OU

P T

IME

DE

LA

Y (

ms

)

PE

RC

EN

TA

GE

RE

LA

TIV

E E

RR

OR

( %

)

α-Chapman’s electronic density profile

For the Chapman’s model:•Δterror does not exceed 0.5% when the frequencies are relatively low.•The error increases when the ray penetrates deep in the ionospheric plasma since numerical integration meets the discontinuity in the refractive index next to the critical frequency.

Figure 1

1313

50

75

100

125

150

175

200

225

250

275

300

325

350

375

0.01 0.1 1 10

fN ( MHz ) [ log ]

h (

km )

Discrete electronic density profile

Discrete electronic density profile defined by parameters as the critical frequency and the height of maximum electronic density.

Figure 2

1414

1

2

3

4

5

6

7

8

9

10

11

12

13

3 6 9 12 15 18 21 24 27 300

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

BREIT&TUVE ( virtual tvirt

) - 18 degrees3D RAY-TRACING ( calculated t

calc ) - 18 degrees

terror

( % ) = tcalc

- tvirt

/ tvirt

- 18 degrees

f ( MHz )

GR

OU

P T

IME

DE

LA

Y (

ms

)

PE

RC

EN

TA

GE

RE

LA

TIV

E E

RR

OR

( %

)

Figure 3a

1515

1

2

3

4

5

6

7

8

9

3 6 9 12 15 18 21 24 27 300

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

BREIT&TUVE ( virtual tvirt

) - 30 degrees3D RAY-TRACING ( calculated t

calc ) - 30 degrees

terror

( % ) = tcalc

- tvirt

/ tvirt

- 30 degrees

f ( MHz )

GR

OU

P T

IME

DE

LA

Y (

ms

)

PE

RC

EN

TA

GE

RE

LA

TIV

E E

RR

OR

( %

)

Figure 3b

1616

1

2

3

4

5

6

3 6 9 12 15 18 21 24 27 300

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

BREIT&TUVE ( virtual tvirt

) - 45 degrees3D RAY-TRACING ( calculated t

calc ) - 45 degrees

terror

( % ) = tcalc

- tvirt

/ tvirt

- 45 degrees

f ( MHz )

GR

OU

P T

IME

DE

LAY

( m

s )

PE

RC

EN

TA

GE

RE

LAT

IVE

ER

RO

R (

% )

For the discrete model:the general trend is analogous to the Chapman’s model except for a more irregular behavior due to the discrete values in the electronic density profile with step of 0.5 km.

Figure 3c

1717

Appendix: checking virtual vertical (oblique) height.

A further analysis has been performed checking the virtual height of the perfect reflector calculatedfirst by a pure geometrical relation (2)and then by means of the equations (3)-(4).

The percentage errors between the two values were always less than 1%.

'0( ) ( , )

r

o

h

v v g vhh f h n h f dh

ng is the group refractive index,h0 the initial height of the ionospherehr the real height of the vertical path

(3)

1 22 2( , ) 1 ( )g v N vn h f f h f

fN is the plasma frequency which has a value about 9√ N with N indicating the electronic density in electrons/cm3

(4)

If secant law is employed,as assured by Martyn’s theorem,the height h’v (hob ) of the reflection point C for the vertical (oblique) path can be derived for each frequency fv (fob)

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CONCLUSIONS

We employed a simple method to test a 3D ray tracing algorithm.

A perfect reflector placed at an altitude equal to the virtual height of reflection was simulated to calculate the virtual time delay that is assumed as a measured time delay. We find a very small percentage relative error between the calculated and the virtual delays, under the hypothesis of a monotonic electronic density profile,for both analytic and numerical models.

Such errors are mainly due to the discrete step of the numerical integration of the differential equations of the ray tracing algorithm and to numerical electronic density profile.

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REFERENCES

Bianchi, C., Settimi,A., Scotto, C., Azzarone, A., Lozito, A.,"A method to test HF ray tracing algorithm in the ionosphere by means of the virtual time delay", J. Adv. Space Res., 48 (10), 1600-1605 (2011); ibid., arXiv:1104.2248v2 [physics.geo-ph] (2011).

Bianchi, C., Settimi, A., Azzarone, A.,IONORT - Ionosphere Ray-Tracing (Programma di ray-tracing nel magnetoplasma ionosferico), INGV Technical Report N. 161, INGV Printing Office, Rome, Italy, 2010.

Davies, K.,Ionospheric Radio, Peter Peregrinus Ltd., London, UK, 1990.

Jones, R. M., Stephenson, J. J.,A versatile three-dimensional ray tracing computer programfor radio waves in the ionosphere,OT Report, 75-76, U. S. Department of Commerce, Office of Telecommunication, U. S. Government Printing Office, Washington, U. S., 1974.