EFFECTIVE

EARTH RADII

FOR

RADAR (OR RF)

AND

OPTICAL

REFRACTION

IN THE

TROPO[SPHERE]

A. Effective Earth Radius Factor (\(K\)) for Radar (or RF) Refraction in the Tropo[sphere]

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.

G. A. Robertshaw (1983)

The MITRE Corporation produced a technical report for the United States Air Force in 1983 addressing the refraction of radio waves in the tropo[sphere] at altitudes above \(1\,\textrm{km}\). The report title page states [that the document is] Approved for public release; distribution unlimited.1 The REPORT DOCUMENTATION PAGE states the REPORT SECURITY CLASSIFICATION as Unclassified.2 The document states:

[…] If the refractivity gradient is constant and vertical everywhere, which corresponds to stratified atmosphere in which the refractivity decreases linearly with altitude, the elegant and widely used effective earth radius method2[3] (EERM) may be employed to obtain geometrical quantities which characterize the propagation path, e.g., earth grazing angle.4 […]

The work referenced (i.e., Nathanson) in the above excerpt in relation to the EERM uses the term \(1/s\) for curvature, expressing the values as negative magnitudes in footnote [1] of that reference:

[1] The value of \(1/s\) varies with time and location. References [310]5 and [354]6 explain that values for \(1/s\) may vary over the United States from \(-1.25\) to \(-1.45\) for altitudes less than \(1\, \textrm{km}\). Values considerably greater than \(-1.3\) have been reported near England [181].7 It is also pointed out that \(1/s=-4/3\) is an average that is almost independent of frequency up to very high radar frequencies. For the visible region, \(1/s≅-1.20\).8,9

The report provides the scale factor for propagation paths within \(1\) kilometer of the surface as a function of surface refractivity, \(N_s\) exclusively:

For paths within one kilometer of the surface, the correct scale factor is a function of the refractivity at the surface \((N_s)\), which establishes the gradient (see appendix), and under ideal conditions, is given by the convenient expression:3[10]\[K=\left[1-0.04665e^{0.005577N_s}\right]^{-1}\tag{15}\] The often quoted \(K = 4/3\) corresponds to \(N_s≅301\). Surface refractivity typically lies between \(200\) and \(400\, N\textrm{-units}\).11

In the Appendix, the author proposes a semi-empirical atmo[sphere] refractivity model:

The refractivity of the earth’s atmosphere can, on the average, be represented by the model described below, which is taken directly from the Radar Handbook.4[12],13

Note: Further to Nathanson’s reference to Skolnick (reference [354]14 in the above excerpt), Skolnick (in reference to Bean) states the following:

The four-thirds earth approximation has several limitations. It is only an average value and should not be used for other than general computational purposes. The correct value of \(k\) depends upon meteorological conditions. Bean15[15],16[16] found that the average value of \(k\) measured at an altitude of \(1\, \textrm{km}\) varies from \(1.25\) to \(1.45\) over the continental United States during the month of February and from \(1.25\) to \(1.90\) during August. In general, the higher values of \(k\) ocur in the southern part of the country. Burrows and Atwood5[17], state that \(k\) lies between \(\frac{6}{5}\) and \(\frac{4}{3}\) in arctic climates.18

(This website will adopt a mean EERF (\(K\)) of \(4/3\) and a maximum EERF (\(K\)) of \(1.45\) for radar or radio frequency (RF) refraction in the tropo[sphere] (subject to future revision); see SUMMARY below.)

D. H. Newsome (1992)

Finally, the effective earth radii for both radar and optical propagation is included in a major European report concerning the proliferation of weather radars. Under the heading, Effective earth radius for radar and optical propagation, the report states:

4.86 The radio refractive index depends on air density, temperature and vapour pressure (e.g. Bean and Dutton, 1967, also explaining ray tracing with a modified earth radius). The effective earth radius \(R\,'\) is found from the mean vertical gradient of the refractive index \(n\) for the height range of the radar beam. For small elevations the multiplication factor is:\[R\,'/R=1/(1+R/n\,(dn/dh))\tag{5}\]4.87 For optical measurements in the lowest \(4\, \textrm{km}\) of the atmosphere, a choice of \(R\,'=1.15\, R\) is adequate.

4.88 The vertical profile of \(dn / dh\) may vary considerably, depending on the meteorological situation. A well-known extreme is the occurrence of very large values of \(R\,'/ R\), resulting in anomalous propagation. For the present purpose it is preferred to use one average \(R\,'\) for every radar. This value will depend on the climatological situation over the observing area and also on the height (above m.s.l.) of the radar beam. Table 11 may serve as a guide for the choice of \(R\,'\). The values shown are for some typical heights and for three (average) temperature profiles. For a certain observation height and temperature the best choice for \(R\,'/R\) will be somewhere between the values specified for the very humid and the very dry atmospheres, tabulated in the two sections of Table 11.19,20,21

[...]

The maximum value for \(R\,'/R\) (i.e., \(k\)) in Newsome’s Table 11 is \(1.45\) under the following conditions: Height (km, m.s.l.) = 0-1; temp. at 0 km = 20°C; R.H. = 100%.22

Summary

Based on the reports of G. A. Robertshaw (1983)⁠—including the above note in reference to M. I. Skolnick (1962 and 1980)⁠—and D. H. Newsome (1992) described above, as well as the earlier work of Schelleng, Burrows, and Ferrell (1933) and Bean and Dutton (1966) described on our web page titled, Radar (or RF) Refraction Curvature in the Tropo[sphere], this website will adopt a mean EERF (\(K\)) of \(4/3\) and a maximum EERF (\(K\)) of \(1.45\) for radar or radio frequency (RF) refraction in the tropo[sphere]. The values of \(4/3\) and \(1.45\) are embedded in the calculators on our web pages titled, (Allegedly Spheroidal) Earth Calculator IV: W/ Mean Tropo[spheric] Radar (or RF) Refraction and (Allegedly Spheroidal) Earth Calculator V: W/ Maximum Tropo[spheric] Radar (or RF) Refraction respectively.

B. Effective Earth Radius Factor (\(K\)) for Optical Refraction in the Tropo[sphere]

Summary

The above reports of G. A. Robertshaw (1983)⁠—in reference to F. E. Nathanson (1969)⁠—and of Newsome (1992), suggest EERF (\(K\)) values of \(1.20\) (or \(5/4\)) and \(1.15\) (or approximately \(7/6\)) respectively. The earlier work of Gauss (1826), the entry in The Encyclopaedia Britannica (1842), and the more contemporary report of Williams and Kahmen (1984), as decribed on our web page titled, Optical Refraction Curvature in the Tropo[sphere] established the (optical) refraction coefficient (\(k\)) values of \(1/7\) (mean) and \(1/5\) (maximum). Under the equation, \[K=\frac{1}{1-k},\] those values translate to an EERF \((K)\) of \(7/6\) (mean) and \(5/4\) (maximum). Hence, this website will adopt a mean EERF (\(K\)) of \(7/6\) and a maximum EERF (\(K\)) of \(5/4\) for optical refraction in the tropo[sphere]. The values of \(7/6\) and \(5/4\) are embedded in the calculators on our web pages titled, (Allegedly Spheroidal) Earth Calculator II: W/ Mean Tropo[spheric] Optical Refraction and (Allegedly Spheroidal) Earth Calulator III: W/ Maximum Tropo[spheric] Optical Refraction respectively.


— FINIS —


The reader is advised to review our web page titled, Purpose and Overview of the (Allegedly Spheroidal) Earth Curvature Analysis, or proceed directly to our web page titled, Introduction to the (Allegedly Spheroidal) Earth Calculators.


  1. G. A. Robertshaw, Effective Earth Radius for Refraction of Radio Waves at Altitudes above 1 Km, Prepared for Deputy for Tactical Systems, Electronic Systems Division, Air Force Systems Command, United States Air Force, Hanscom Air Force Base, Massachusetts, Project No. 6460, Prepared by The MITRE Corporation, Bedford, Massachusetts, Contract No. F19628-82-C-0001, December 1983, DTIC SELECTED JAN 23 1984 (see https://apps.dtic.mil/sti/citations/ADA137095).↩️

  2. Ibid., REPORT DOCUMENTATION PAGE.↩️

  3. Fred E. Nathanson, Radar Design Principles (New York: McGraw-Hill, 1969), pp. 28–33.↩️

  4. G. A. Robertshaw, op. cit., p. 5.↩️

  5. P. Rice, et al., Transmission Loss Predictions for Tropospheric Communication Circuits, Natl. Bureau Std. Tech Note 101, Vols. 1 and 2, 1965 (see https://nvlpubs.nist.gov/nistpubs/Legacy/TN/nbstechnicalnote101-1.pdf and https://nvlpubs.nist.gov/nistpubs/Legacy/TN/nbstechnicalnote101-2.pdf).↩️

  6. M. I. Skolnik, Introduction to Radar Systems (New York: McGraw-Hill, 1962).↩️

  7. I. M. Hunter, and T. B. A. Senior, Experimental Studies of Sea-surface Effects on Low-angle Radars, Proc. IEE, Vol. 113, No. 11, November 1966, pp. 1731–1740.↩️

  8. Fred E. Nathanson, op. cit., pp. 29–30.↩️

  9. The negative \(1/s\) curvature values are expressed as positive values in reference to effective earth radius factors, i.e., \(K\) values.↩️

  10. W. C. Jakes, ed., Microwave Mobile Communications (New York: John Wiley and Sons, 1974), pp. 84–85.↩️

  11. G. A. Robertshaw, op. cit., p. 13. Note: In contrast to the original document, letters designating variables or mathematical symbols in the above excerpt have been italicized in accordance with the normal convention. This applies to \(N _s\), \(K\), \(e\), and \(N\)-units.↩️

  12. M. I. Skolnik, Radar Handbook (New York: McGraw-Hill, 1970), p. 24–10.↩️

  13. G. A. Robertshaw, op. cit., APPENDIX – SEMI-EMPIRICAL ATMOSPHERIC REFRACTIVITY MODEL, p. 33–38.↩️

  14. M. I. Skolnik, loc. cit.↩️

  15. B. R. Bean, The Geographical and Height Distribution of the Gradient of Refractive Index, Proc. IRE, Vol. 41, April 1953, pp. 549–550.↩️

  16. B. R. Bean et al., “A World Atlas of Atmospheric Radio Refractivity,” U.S. Dept. of Commerce, ESSA Monograph 1, 1966.↩️

  17. C. R. Burrows and S. S. Attwood, Radio Wave Propagation (New York: Academic Press, Inc., 1949).↩️

  18. M. I. Skolnik, Introduction to Radar Systems, Second Edition (New York: McGraw-Hill, 1980), p. 450.↩️

  19. D. H. Newsome, Weather Radar Networking – Cost 73 / Final Report, Commission of the European Communities (Dordrecht: Kluwer, 1992), p. 45 (see https://op.europa.eu/en/publication-detail/-/publication/5edab5a3-b84e-4713-b220-f1afc519ff14). Note: In contrast to the original document, letters designating variables in the above excerpt have been italicized in accordance with the normal convention. This applies to \(R\) and \(R\,'\), \(η\), \(n\), and \(h\).↩️

  20. Ibid., Table 11 Factors \(R\,'/R\) (radar wavelengths, wet-adiabatic temperature profiles), p. 46.↩️

  21. 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).↩️

  22. D. H. Newsome, op. cit., p. 46.↩️


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