SOCIETÁ ITALIANA DI FISICA
XCVII CONGRESSO NAZIONALE
L’AQUILA, 26-30 SETTEMBRE, 2011
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
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
08/04/2015
1
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 profile
for both analytic and numerical models.
2
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.
3
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).
4
3D ray tracing algorithm: calculated group time delay.

1  c2
H (r , ,  , t , kr , k , k ,  )  Re  2  kr 2  k 2  k 2   n 2 
2 

The Hamiltonian H
is a propagation constant
during the ray path.
dr H

d kr
d 1 H

d r k
d
1

d r sin 
dt
H

d

dkr
H
d
d

 k
 k sin 
d
r
d
d
dk 1 H
dr
d
 (
 k
 k r cos 
)
d r 
d
d
1
H
dr
d
H dk

(
 k sin 
 k r cos 
)
r sin  
d
d
k d
d H

d
t
The independent variable τ
is suitably chosen as
a parameter satisfying
the monotonocity condition
along the ray path.
  ct
5
The 3D ray tracing algorithm ends the integration when the ray reaches the ground after reflection.
  ct
tcalc   ti
i
(1)
6
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).
7
The measured group time delay treal can be compared to the virtual time delay tvirt.
This last quantity can be obtained immediately
using 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.
8
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 calculated
group 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.
9
Secant law
fob is the oblique frequency,
fv is the vertical frequency
and φ is the incidence angle.
fob  fv  sec 
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:
hob ( fob )  h'v ( fv )
Breit&Tuve’s theorem
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
tcalc  tvirt
10
The virtual time delay tvirt can be immediately obtained,
using a pure geometrical relation:
tvirt
2d

c
(2)
11
Analytic and numerical electronic density profiles:
percentage relative error of delay.
α-Chapman’s electronic density profile
Figure 1
BREIT&TUVE ( virtual tvirt ) - 18 degrees
//
- 30 degrees
//
- 45 degrees
3D RAY-TRACING ( calculated tcalc ) - 18 degrees
//
- 30 degrees
//
- 45 degrees
terror ( % ) = tcalc - tvirt / tvirt - 18 degrees
//
- 30 degrees
//
- 45 degrees
12
3.0
11
GROUP TIME DELAY ( ms )
9
2.0
8
7
1.5
6
5
1.0
4
3
PERCENTAGE RELATIVE ERROR ( % )
2.5
10
0.5
2
1
3
6
9
12
15
18
21
24
27
0
30
f ( MHz )
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.
12
Discrete electronic density profile
375
Figure 2
350
325
300
275
h ( km )
250
225
200
175
150
125
100
75
50
0.01
0.1
1
10
fN ( MHz ) [ log ]
Discrete electronic density profile defined by parameters
as the critical frequency and the height of maximum electronic density.
13
Figure 3a
BREIT&TUVE ( virtual tvirt ) - 18 degrees
3D RAY-TRACING ( calculated tcalc ) - 18 degrees
terror ( % ) = tcalc - tvirt / tvirt - 18 degrees
13
4.5
12
4.0
3.5
GROUP TIME DELAY ( ms )
10
9
3.0
8
2.5
7
2.0
6
5
1.5
4
1.0
PERCENTAGE RELATIVE ERROR ( % )
11
3
0.5
2
1
3
6
9
12
15
18
21
24
27
0
30
f ( MHz )
14
Figure 3b
9
4.5
8
4.0
3.5
GROUP TIME DELAY ( ms )
7
3.0
6
2.5
5
2.0
4
1.5
3
1.0
2
1
PERCENTAGE RELATIVE ERROR ( % )
BREIT&TUVE ( virtual tvirt ) - 30 degrees
3D RAY-TRACING ( calculated tcalc ) - 30 degrees
terror ( % ) = tcalc - tvirt / tvirt - 30 degrees
0.5
3
6
9
12
15
18
21
24
27
0
30
f ( MHz )
15
Figure 3c
BREIT&TUVE ( virtual tvirt ) - 45 degrees
3D RAY-TRACING ( calculated tcalc ) - 45 degrees
terror ( % ) = tcalc - tvirt / tvirt - 45 degrees
6
5.0
GROUP TIME DELAY ( ms )
5
4.0
3.5
4
3.0
2.5
3
2.0
1.5
2
1.0
PERCENTAGE RELATIVE ERROR ( % )
4.5
0.5
1
3
6
9
12
15
18
21
24
27
0
30
f ( MHz )
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.
16
Appendix: checking virtual vertical (oblique) height.
A further analysis has been performed checking
the virtual height of the perfect reflector calculated
first 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%.
(3)
hr
h v ( fv )  h0   ng (h, fv )dh
'
ho
(4)
ng (h, f v )  1  f N2 (h) f v2 
1 2
ng is the group refractive index,
h0 the initial height of the ionosphere
hr the real height of the vertical path
fN is the plasma frequency
which has a value about 9√ N
with N indicating the electronic density
in electrons/cm3
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)
17
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 program
for 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.
19