A Reflection Theory for Propagation 

 Beyond the Horizon* 



By H. T. FRIIS, A. B. CRAWFORD and D. C. HOGG 



(Manuscript received January 9, 1957) 



Propagation of short radio waves beyond the horizon is discussed in terms 

 of reflection from layers in the atmosphere formed by relatively sharp gra- 

 dients of refractive index. The atmosphere is assumed to contain many such 

 layers of limited dimensions with random position and orientation. On this 

 basis, the dependence of the propagation on path length, antenna size and 

 ivavclength is obtained. 



INTRODUCTION 



It was pointed out several years ago^ that power propagated beyond 

 the radio horizon at very short wavelengths greatly exceeds the power 

 calculated for diffraction around the earth. This beyond-the-horizon 

 propagation has stimulated numerous experimental and theoretical in- 

 vestigations.- Booker and Gordon,^ 'S'illars and Weisskopf^ and others 

 have developed theories based on scattering of the radio waves by tur- 

 bulent regions in the troposphere. This paper proposes a theory in which 

 uncorrelated reflections from layers in the troposphere are assumed re- 

 sponsible for the power propagated beyond the horizon. 



In developing this theory, some arbitrary assumptions of necessity 

 have been made concerning the reflecting laj^ers since, at the present 

 time, our detailed knowledge of the atmosphere is insufficient. However, 

 calculations based on the theory are found to be in good agreement with 

 reported measurements of beyond-the-horizon propagation. 



Measurement of the dielectric constant of the atmosphere^ has shown 

 that relatively sharp variations in the gradients of refractive index exist 

 in both the horizontal and vertical planes. Although the geometrical 

 structure of the boundaries formed by the gradients is not well knoATO, 

 one may postulate an atmosphere of many layers of limited extent and 



* This material was presented at tlie I.R.E. Canadian Convention, Toronto, 

 Canada, October 3, 1956. 



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