# Distance Measurement with Electromagnetic Wave Dispersion

## Abstract

An atmosphere above a half space considered with the dispersion represented by a relation between the electric field and the induction which contains derivatives of rational order and is similar to the empirical formula of Cole and (1941), commonly used in experimental physics, and to the formula used by (1991) in studying the dispersion of energy in electric networks. The dispersion of a monochromatic wave is modelled considering the index of refraction n as a rational function of a rational power of the imaginary frequency *if*, as usually in geophysics, and is a polymorphic function of *f*; this function, for each frequency, gives a set of different velocity fields, whose number depends on the rational exponent of *if*. Each electromagnetic wave leaving the source, with given *f* and direction, is split in a number of waves with different velocities; if *n* is a function of position, the paths of the waves are different and reach a given elevation at different points and times. If *n* is independent of the position, the paths of the waves coincide although the waves have different velocities. The length of a path and the travel time of electromagnetic waves in the atmosphere of a flat Earth model are computed. It is found that the difference between the arc length of the ray and the chord is nil to the second order of refractivity. It is also seen that a change of water content is layers of the atmosphere, leaving the average velocity profile to a given elevation unchanged, may change the length of the ray paths to the elevation. It is found that the separation of the rays with the same frequency and direction at the source, causes small uncertainties in electromagnetic distance measurements which increase with the frequency. In the (1985) atmospheric model we considered frequencies in the range 1 GHz to 2 GHz and found that the arrival of the phases of the rays, with the same frequency in this range, with a zenithal angle smaller that 27π/5 and that a distance of about 10^{4} km, are spread in less than 0.01 ns or 0.3 cm; which does not influence the accuracy presently achieved in distance measurements with electromagnetic waves. The dissipation of energy of the rays in the atmospheric model used, for zenithal angles smaller than 27π/5, is negligible for any length of the path. Formulae are given for the retrieval of a spherical model of the atmosphere of the Earth from a set of differences of the times of arrival, at two observing stations, of the waves emitted from satellites of known orbits.

## Key words

Atmosphere electromagnetic waves ray-path splitting distance measurements## Preview

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## References

- Allnutt, J. E. (1989).
*Satellite to ground radiowave propagation*. Peter Peregrinus Limited Press.Google Scholar - Antonelli, P. and Caputo, M. (1994). Determinazione di un modello dell’atmosfera dai tempi di volo ai satelliti:311-318. In
*Atti XIII Convegno Nazionale Gruppo Naz. Geo fis*. Terra Solida, pages 311–318, CNR Roma.Google Scholar - Bagley, R.L. and Torvik, P. J. (1986). On fractional calculus model of viscoelastic behavior.
*J. of Rheology*, 30(1):133–155.CrossRefGoogle Scholar - Caputo, M. (1989). The rheology of an anaelastic medium studied by means of the observation of the splitting of its eigenfrequencies.
*J. Acoust. Soc. Am.*, 86(5):1984–1987.CrossRefGoogle Scholar - Caputo, M. (1993a). Free modes splitting and alteration of elettrochemically polarizable media.
*Atti Accad. Naz. Lincei, Rend. Fis.*, 9(4):89–98.CrossRefGoogle Scholar - Caputo, M. (1993b). The splitting of the seismic rays due to the dispersion in the Earth’s interior.
*Atti Accad. Naz. Lincei, Rend. Fis.*, 9(4):279–286.CrossRefGoogle Scholar - Caputo, M. and Mainardi, F. (1971). Linear models of dissipation in anaelastic solids.
*Rivista del Nuovo Cimento*, 2(1):161–198.CrossRefGoogle Scholar - Cole, K.S. and Cole, R. H. (1941). Dispersion and absorbtion in dielectrics.
*J. of Chemical Physics*, 9:341–349.CrossRefGoogle Scholar - Debye, P. (1928).
*Polar molecules*. Chemical catalogue company, New York.Google Scholar - Dziewonski, A. M., Hales, A.L., and Lapwood, E. R. (1975). Parametrically simple Earth models consistent with geophysical data.
*Phys. Earth. and Plan. Int.*, 10:12–25.CrossRefGoogle Scholar - Jacquelin, J. (1991). A number of models for CPA impendances of conductors and for relaxation in non-Debye dielectrics.
*J. of Non-Crystalline Solids*, 131(1331):1080–1083.CrossRefGoogle Scholar - Körnik, H. and Müller, G. (1989). Rheological models and interpretation of postglacial uplift.
*Geo-phys. J. R. Astr. Soc.*, 98:245–253.Google Scholar - Liebe, H.J. (1985). An updated model for millimeter wave propagation in moist air.
*Radio Science*, 20(5):1069–1089.CrossRefGoogle Scholar - Ruggiero, V. and Caputo, M. (1998). Tomography of the atmosphere for times of flight to satellite observed from one and two stations.
*π Nuovo Cimento (C)*, 21(2):177–187.Google Scholar - Slichter, L.B. (1932). The theory of the interpretation of seismic Travel-Time curves in horizontal structures.
*Physics*, 3:273–295.CrossRefGoogle Scholar