1. 21/05/2015 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 1 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

2. 2 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.

3. 3 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.

4. 4 Introduction In case of a real measurement, it is possible to compare the calculated group time delay t calc with the measured time delay t real. 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 t real 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).

5. 5 3D ray tracing algorithm: calculated group time delay. 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.

6. 6 The 3D ray tracing algorithm ends the integration when the ray reaches the ground after reflection. (1)

7. 7 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).

8. 8 The measured group time delay t real can be compared to the virtual time delay t virt. 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 t real = t virt, a monotonic increasing trend of the electronic density profile is also required otherwise the valley between E and F layers invalidates the relation.

9. 9 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 t calc. 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.

10. 10 Secant law f ob is the oblique frequency, f v is the vertical frequency and φ is the incidence angle. Martyn’s theorem The oblique virtual height h ob of reflection at the oblique frequency f ob is equal to the vertical virtual height h’ v of reflection at the vertical frequency f v, according to the secant law: Breit&Tuve’s theorem The calculated group time delay t calc of the wave propagating in the effective path at group velocity compatible with the refractive index is equal to the virtual time delay t virt of the wave propagating along the oblique virtual path at the speed c of light in vacuum

11. 11 The virtual time delay t virt can be immediately obtained, using a pure geometrical relation: (2)

12. 12 Analytic and numerical electronic density profiles: percentage relative error of delay. α-Chapman’s electronic density profile For the Chapman’s model: Δt error 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

13. 13 Discrete electronic density profile Discrete electronic density profile defined by parameters as the critical frequency and the height of maximum electronic density. Figure 2

14. 14 Figure 3a

15. 15 Figure 3b

16. 16 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

17. 17 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%. n g is the group refractive index, h 0 the initial height of the ionosphere h r the real height of the vertical path (3) f N is the plasma frequency which has a value about 9√ N with N indicating the electronic density in electrons/cm 3 (4) If secant law is employed, as assured by Martyn’s theorem, the height h’ v ( h ob ) of the reflection point C for the vertical (oblique) path can be derived for each frequency f v ( f ob )

18. 18 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.

19. 19 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.