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Ok in my last post I made a feeble attempt to write some code. Without the full math understanding. So I will attempt this again but I will add a new image, as seen on the right,  that will help you out.  Where R is the Slant Range, Where H is the Aircrafts altitude from the antenna and ε is the angle between R and Topogar or the distance between the antenna and the aircraft projected on the ground. R, H, and Topogar will always form a right triangle. And simply we must use a Great Circle Calculation to determine Topogar. If the station Slant Range (R) is greater than the line of sight you will not hear the Nav Aid.

In the NavAids published info Latitude and Longitude the antenna is described as so many feet off the ground we will call it ha. So we must add the antenna height, or ha to get the distance been the antenna and center of the Earth. While Re is a vector from the center of the Earth to sea level, and then is extended aircraft's altitude above sea level.

A second way of looking at this is that you have 3 triangles:

• made of the following sides: R and Topogar and the Altitude of the aircraft to the center of the antenna.
• made of the following sides: (1) the antenna, latitude, and longitude and the height of the antenna to the center of the earth. (2) The center of the earth to the surface.  (3) The curved section of the earth.
• made of the following sides: (1) the antenna, latitude, and longitude and the height of the antenna above sea level. (2) Slant Range a line between the antenna and the aircraft. (3) a line from sea level to the aircraft.

Now Back to the math, what I have shown was radar slant range! The formula for computing DME  slant range is the following:

 D  = √((re+ha)2-re2)) + √(re + h)       can now be reformatted as:     D  = √( ha2+2reha) + √( hs2 + 2)     ha = 0  Assuming a receiver on the sea surface     D  ≈ √(2rehs) SITE: Owen, J. I. R., “Results of an Investigation into the use of 1175 MH and 1202 MHz for GNSS Signals in European Airspace,” Proceedings of the 2002 National Technical Meeting of The Institute of Navigation, SanDiego, CA, January 2002, pp. 742–749