NavList:

Robert Lafleur, you asked:
"Can somebody explain why in Bowditch the formula for distance to the horizon is D = 1.169 x sqr root of h
and elsewhere on the internet it is D = 1.22459 x sqr root of h
Where: D is in nautical miles and h is height of eye in feet"

Short answer: The Bowditch factor is generally accurate for height of eye in feet yielding distance in nautical miles. Use it!

The factor of about 1.16 or 1.17 is correct under two important conditions: distance is measured in nautical miles, and refraction of light exists (it does!) and is not extreme (due to unusual weather conditions). The factor of 1.224 applies for distances in statute miles and assumes an unrealistic case where light does not refract. The first issue is not necessarily a problem so long as we use it correctly. But of course most distances in marine circumstances are measured in nautical miles so generally that's preferred --a better choice. There's really no excuse for the second assumption. Light rays do not travel in straight lines in the real atmosphere. They are bent by refraction. Although the exact amount of refraction is variable and depends on the density profile of the air in the lower atmosphere, the average is stable and well-known and has been calculated and studied for over 250 years. Refraction increases the apparent distance to the horizon by about 8%. This is directly incorporated in the factor in the standard Bowditch formula.

Some things to keep in mind:

• Refraction is normally a little variable so you should trust this equation to about 1% under normal conditions. So if you someone tells you to calculate the distance to the horizon for a height of eye of 3 feet, for example, and then offers up the square root of 3 to eight decimal places, stop right there. You only need to work to +/-1% on the inputs and intermediate calculations, and you should only trust the result to +/-1%.
• Refraction can be exotic, extreme. If you're in unusual weather conditions, for example, in arctic waters with calm air, you may experience unusual refraction which can over-rule these simple relationships.
• Height of eye in feet can be tricky to measure on a moving vessel. As you rise up and down on ocean swells, your height of eye can easily change by several feet with corresponding changes in the distance to the horizon. And do you really think you can measure your height of eye to exactly 3.00000 feet? Of course not. So use a little common sense. Maybe try a couple of different approximate values for height of eye to see how much they affect the calculation of distance to the horizon.
• The distant horizon that we see is composed of the overlapping crests of waves. Therefore your height should be relative to typical wave-top heights. If you're in open water, well away from land, you can safely assume that wave conditions near your vessel are more or less identical to wave conditions some miles away from you. It is therefore reasonable to look over the side and estimate how high you are above local wave-tops. Use that as your height of eye.
• This calculation does not depend on ocean tide height or the height of the body of water above sea level. If you're sailing in the Bay of Fundy at high tide, then height above wave-tops in feet is what you put into the equation, as in any other case. If you're sailing on Lake Tahoe (in a boat small enough to see a horizon before the land), then what counts is your height of eye above the lake's wave-tops in feet, as in any other case. Refraction does decrease modestly at very high altitudes, and there should be an adjustment there.
• The horizon is not a sharp line. It has "depth". If you see an object in the distance seemingly floating on the water "right at the horizon", the range to that object can be the calculated distance to the horizon plus or minus a mile or so --because of small variations in height (on waves) "out there". This depth is distinct from the +/- a percent I mentioned above and usually larger, too.

Please let us know if you have any other questions on this.

Frank Reed
Clockwork Mapping / ReedNavigation.com
Conanicut Island USA