RADAR (OR RF)

REFRACTION

CURVATURE

IN THE

TROPO[SPHERE]

 
 

Radar (or Radio Frequency) Refraction Curvature in the Tropo[sphere]: The Radar (or Radio Frequency) Coefficient of Refraction \((k)\) and the Effective Earth Radius Model (EERM)

Nota Bene

As stated in the first paragraph of our web page titled, Refraction Overview:

The purpose of the REFRACTION section is to (a) present to the reader, the underlying rationale for (spheroidal earth) geodesists using (alleged) tropo[spheric] refraction to explain (in many cases) the visibilty of subjects of observation that should otherwise be below the horizon (i.e., below and normal to the observer’s horizon line of sight), and (b) give the benefit of the doubt to the fallacious spheroidal earth paradigm by allowing readers to assume the impact of maximum tropo[spheric] refraction on their field observation calculations (see Introduction to the (Allegedly Spheroidal) Earth Calculators).

Such calculations are of paramount importance in REVEALING THE PLANAR LARGE-SCALE STRUCTURE OF THE EARTH’S SURFACE.

Introduction

Having established the historic mean and maximum values for optical refraction curvature in the tropo[sphere] to be \(1/7\) and \(1/5\) respectively, of the (alleged) curvature of the earth,1 a digression to radar refraction curvature in the tropo[sphere] is necessary in order to exploit a tropo[spheric] refraction model developed for radar operations, the essential concept of which is applicable to tropo[spheric] refraction in the optical range.

Schelleng, Burrows, and Ferrell (1933)

It was shown by Schelleng, Burrows, and Ferrell2 that the determination of the magnitude of refraction of ultra-short radio waves by the lower atmosphere could be significantly simplified by assuming an earth with a larger radius such that the curvature of the refracted radio wave matches this new, fictitious (alleged) curvature of the earth, effectively allowing the refracted radio wave to be treated analytically as a straight ray.

The following excerpt from their paper is pertinent:

    [...]
    The radius of curvature of a ray traveling horizontally in the lower atmosphere can be readily calculated if it is known how the refractive index, \(n\), varies from point to point. If \(H\) is the altitude above sea level, the radius of curvature of the ray is simply
\[ρ=-\frac {n} {dn/dH}.\]
But since \(n=\sqrt{ε}\), where \(ε\) is the dielectric constant, the radius of curvature is
\[ρ=-\frac {2} {dε/dH},\]
provided \(n\) is not very different from unity.
    In Appendix II the estimation of this radius of curvature is discussed in some detail. While some of the data upon which such a calculation can be based are rather uncertain, it appears that a good approximation is obtained by assuming the radius of curvature, \(ρ\), of the refracted ray to be four times the radius of the earth, \(r\small_{0}\). As pointed out in the appendix, this varies to some extent with weather, and even as an average value, it may have to be changed when more reliable data on dielectric constants become available.
    On first consideration of the ways in which refraction can be taken into account, it appears that the attempt must complicate an already involved situation. Fortunately, however, refraction is much simpler to calculate than diffraction or reflection. The method is presented rigorously in Appendix III. At this point we shall merely state the result and show its plausibility.
    In ultra-short wave work we are almost always concerned with propagation in a nearly horizontal direction. The curvature of the ray is \(1/ρ\), while that of the earth is \(1/r\small_{0}\). We are interested, however, in the relative curvature, which we shall call \(1/r_e\). If, instead of using simple rectangular coordinates, we transform to a coordinate system in which the ray is a straight line, the curvature of the earth will become \(1/r_e\), which is \(1/r\small_{0}-1/ρ\). The equivalent radius of the earth would be\[r_e=r\small_{0}\left(\dfrac{1}{1-r\small_{0}/ρ}\right),\] and is therefore greater than the actual radius of the earth by the factor \(\frac{1}{1-1/4}\) which is \(1.33\,\textrm{[or}\,4/3\textrm{]}\). This fictitious radius is therefore \(8500\, \textrm{km}\) instead of \(6370\,\textrm{km}\).3 [...]

Hence, from the above posit, whereas the refraction coefficient \(k\) corresponds to the reciprocal of the radius of curvature \(ρ\) of the refracted ray, then \(k_{mean}=1/4=0.25\) in the radar or microwave range of the electromagnetic spectrum.

Bean and Dutton (1966)

Bean and Dutton4 extoll the essential practicality of the (linear) effective earth radius model advanced by Schelleng et al., but propose modifications to more realistically address the actual height distribution of refractive index in the atmo[sphere]. Specifically, they state:

[…] This method of accounting for atmospheric refraction permits a tremendous simplification in the many practical problems of radio propagation engineering although the height distribution of refractive index implied by this method is not a very realistic representation of the average refractive index structure of the atmosphere. This section will consider the refractive index structure assumed by the effective earth’s radius model and how this differs from the observed refractive index structure of the atmosphere. Further, the limits of applicability of the effective earth’s radius approach will be explored and a physically more realistic model, the exponential, will be described for those conditions where the effective earth’s radius model is most in error.5

They further state, however,

One might wonder […] [in reference to data supporting the above limitation] […] why the effective earth’s radius model has served so well for so many years. It appears that this success is due to the \(4/3\) earth model being in essential agreement with the average \(N\) structure near the earth’s surface which largely controls the refraction of radio rays at the small values of \(θ\small_{0}\) common to troposphere communication system.6

(The term \(θ\small_{0}\) is the apparent elevation angle to the radar illuminated target as opposed to the true elevation angle.)7

But based upon a review of higher elevation \(N\) profile conditions in different seasons and climates, Bean and Dutton propose a height-dependent exponential function:

… it is seen that the data strongly suggest that \(N\) may be represented by an exponential function of height of the form:\[N(h)=N\small_{0}\,\textrm{exp}\{-bh\},\] in the altitude range of \(1\) to \(9\,\textrm{km}\) above sea level.8

The height dependence of \(N\) is further examined on our web page titled, Effective Earth Radii for Radar (or RF) and Optical Refraction in the Tropo[sphere] .

Depiction of the Effective Earth Radius Model (EERM)

P RE (Refracted earth horizon) P GREE (Effective earth geometric and refracted horizon) P EE (φ, λ, H EE ) (Effective earth orthometric height position of radar or radio frequency transceiver) P E (φ, λ, H E ) (Orthometric height position of radar or radio frequency transceiver) P EE (φ, λ) (Effective earth sea level position of radar or radio frequency transceiver) O (Center of the allegedly spheroidal earth) C E (Alleged circumference of the of the allegedly spheroidal earth) C EE (Alleged circumference of the allegedly spheroidal effective earth) H EE H E P GE (Geometric earth horizon) P E (φ, λ) (Sea level position of radar or radio frequency transceiver) Radar transmitter or radio frequency transceiver's (unrefracted) horizon line of sight Radar transmitter or radio frequency transceiver's (refracted) horizon line of sight Radar transmitter or radio frequency transmitter's effective earth horizon line of sight R 1 R 1 KR 1

Figure 1. Depiction of curved refracted ray over earth of radius \(R_1\)9 (solid magenta ray in lower part of disgram) and (effective) straight ray over earth of radius \(KR_1\) (dotted magenta ray in upper part of diagram).10,11 (Note: Object sizes and distances have been scaled or exaggerated for illustrative purposes.)

In the lower part of Figure 1 (illustrated by solid lines), point \(P\small_{E}\,(φ,λ)\)12 represents the (nominal) sea level position of the radar or radio frequency transceiver whereas point \(P\small_{E}\,(φ,λ,H\small_{E})\) is the (actual) orthometric height position of the radar or radio frequency transceiver. The radar transmitter or radio frequency tranceiver’s unrefracted horizon line of sight is tangent to the (alleged) circumference \(C\small_{E}\) of the (allegedly spheroidal) earth (of alleged radius \(R_1\)) at the geometric earth horizon represented by point \(P\small_{GE}\) whereas the refracted horizon line of sight is tangent to the (alleged) circumference \(C\small_{E}\) of the (allegedly spheroidal) earth (again, of alleged radius \(R_1\)) at the refracted earth horizon represented by point \(P\small_{RE}\).

In the upper part of Figure 1 (illustrated by dashed lines), point \(P\small_{EE}\,(φ,λ)\) represents the effective earth (nominal) sea level position of the radar or radio frequency transceiver whereas point \(P\small_{EE}\,(φ,λ,H\small_{EE})\) is the effective earth (actual) orthometric height position of the radar or radio frequency transceiver. The radar transmitter or radio frequency tranceiver’s refracted horizon line of sight is now tangent to the (alleged) circumference \(C\small_{EE}\) of the (allegedly spheroidal) earth (of alleged radius \(KR_1\)) at the effective earth geometric and refracted horizon represented by point \(P\small_{GREE}\), the geometric and refracted horizons now being coincident.


— FINIS —


The reader is advised to proceed to the Effective Earth Radii for Radar (or RF) and Optical Refraction in the Tropo[sphere] subsection that provides a literature review concerning limitations to the effective earth radius model (and the necessary incorporation of modifications thereto) as well as literature implying its conceptual adaptation to the optical range, concluding with the provision of effective earth radii for both radar (or RF) and optical refraction in the tropo[sphere], the corresponding \(K\) values being applied in the CALCULATORS section.


  1. See our web page titled, Optical Refraction Curvature in the Tropo[sphere]: The Optical Coefficient of Refraction (\(k\)).↩️

  2. J.C. Schelleng, C.R. Burrows, and E.B. Ferrell. “Ultra-short wave propagation.” The Bell System Technical Journal, 12(2), April 1933, pp. 125–161. (Also published in Proceedings of the Institute of Radio Engineers, 21(3), March 1933, pp. 427–463.)↩️

  3. Ibid., pp. 139–140. It should be noted that the lower-case Greek letter \(ρ\) (rho) used to designate the radius of curvature of the ray, is also used on our web page titled, Tropo[spheric] Refraction: The Nature of Tropo[spheric] Refraction to designate air density.↩️

  4. B. R. Bean and E. J. Dutton, Radio Meteorology (U.S. Department of Commerce: National Bureau of Standards Monograph 92 – Issued March 1, 1966.), U.S. Government Printing Office, Washington, D.C., 1966.↩️

  5. Ibid., p. 56.↩️

  6. Ibid., p. 60. It should be noted that \(N\) (also designated as \(N_s\)) is known as the surface refractivity and is defined as \[N_s=(n-1)\times10^6,\] where \(n\) is the index of refraction (from W. C. Jakes, ed., Microwave Mobile Communications (New York: John Wiley and Sons, 1974), equation (2.1-15), p. 85).↩️

  7. Ibid., pp. 49–50.↩️

  8. Ibid., p. 61. Note that \(b\) is a constant.↩️

  9. The designation of the (alleged) radius of the earth as \(R_1\) is taken from the World Geodetic Sytem 1984 (WGS 84) designation of \(R_1\) as the (alleged) mean radius of the (alleged) three semi-axes of the (allegedly) spheroidal earth (see our web page titled, (Allegedly Spheroidal) Earth Mensuration).↩️

  10. Cf. Donald G. Bodnar. “3 THE PROPAGATION PROCESS,” in Principles of Modern Radar, edited by Jerry L. Eaves and Edward K. Reedy (New York: Van Nostrand Reinhold, 1987), especially under the heading, Effective Earth Model, pp. 57–62, and in particular, Figure 3-13 (p. 61) and Figure 3-14 (p. 62), both figures being from C. R. Burrows and S. S. Attwood, Radio Wave Propagation (Academic Press Incorporated: New York, 1949).↩️

  11. The symbol \(K\) refers to the Effective Earth Radius Factor (EERF); see our web page titled, Effective Earth Radii for Radar (or RF) and Optical Refraction in the Tropo[sphere]. In terms of the refraction coefficient \(k\), \[K=\dfrac{1}{1-k}.\] The mean values for \(K\) and \(k\) are \(4/3\) and \(1/4\) respectively for RF radiation.↩️

  12. Where \(φ\) and \(λ\) refer to latitude and longitude respectively.↩️


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DATE 2023-JAN-12 2023-MAR-15 2023-NOV-10