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Quoting from lecture notes of Andrew young at San Diego State University:

For less extreme, but very reliable, observations, consider some listed by Commander C. L. Garner of the Coast and Geodetic Survey in 1933 [...] He also credits the 1911 sighting of the Fairweather Mountains in Alaska [highest point is Mt. Fairweather 15 325 ft or 4670 m] from the ship Explorer from the Gulf of Alaska, 330 miles [531 km] away."

Earth’s mean average radius is 3959 miles by a right angled tangent of 330 miles gives a hypotenuse extending well above the surface of the earth of sqrt(39592 + 3302) = 3973 miles. Subtracting the radius 3959 miles gives the drop in the curvature of the earth from the beginning point where the Explorer made the sighting: 13.7 miles, or 72 492 ft.

72 492 ft minus Mt. Fairweather’s height 15 325 ft gives 57 167 ft to spare for a whole mountain range to disappear beneath the curvature of the earth.

What could account for this sighting 'through' the curve of the earth?

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    $\begingroup$ I'm voting to close this question as off-topic because this has no relation to astronomy. $\endgroup$ – HDE 226868 Sep 13 '15 at 22:28
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    $\begingroup$ Refraction, in this case probably anomalous refraction $\endgroup$ – Conrad Turner Sep 14 '15 at 3:51
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    $\begingroup$ Hi Gerald, this seems to better fit in the Earth Science site, one of their mods has agreed it is more likely to fit in there so I will migrate this over for you now. $\endgroup$ – RhysW Sep 14 '15 at 17:20
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    $\begingroup$ You've missed refraction, which will give a (smallish) range increase, and also that the viewpoint on the ship will not be at sea level, but probably some tens of metres above it. That will give a substantial increase in visual range, but I don't know whether it's enough. $\endgroup$ – Semidiurnal Simon Sep 15 '15 at 19:32
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    $\begingroup$ You also have about an 8% increase is visible distance just due to normal atmospheric refraction. with a drastic temprature change the refraction can get a lot higher. en.wikipedia.org/wiki/Atmospheric_refraction $\endgroup$ – John Mar 16 '17 at 19:55
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I think what was observed is a superior mirage. A superior mirage occurs when warm air overlies a very cold layer of air, aka an inversion. You don't often see a superior mirage in lower latitudes, but they seem to be somewhat common in and near the Arctic regions.

With normal atmospheric conditions and taking into consideration refraction, about the furthest one could see a 15,325 foot mountain from, say, a 100 foot high deck is roughly 177 miles, depending on the tide. The only possible explanation is a very strong superior mirage.

There are a couple of good articles here and here that describe the conditions necessary:

towering image as seen from shore

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The question I'm answering is: Is it physically possible to see mountains from 500 km away?

It is very likely possible to see the top of Mt. Everest from the top of some mountain that is 500+ km distant. The other mountain would have to be at least 10000 feet high and be located 500 - 550 km from Mt Everest, and there has to be no other mountains blocking the view. Much better would be a 20000 ft high mountain because then you could see over 10000 ft high mountains in between

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    $\begingroup$ Could you please justify your statement with some science. A link to a reference where this has been proved would also improve you answer. $\endgroup$ – Fred Mar 16 '17 at 11:17

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