EDUCATOR’S PAGE


And, since “2rE” is a constant, we can pull it out to further simplify: d h(2)(6371 km) d h d h )


The formula above supposes we provide height in kilome- ters, but most people would not exceed a height of 2 meters, and so a conversion can be added into the equation to accom- modate a change to using meters (1 km = 1000 m, or 1 m = 1/1000 km).


dh(m)/(1000 m/km)km) dhh dh


In the final equation immediately preceding, then, “d” is given in kilometers when “h” is in meters. The solutions estimate the distance to the horizon based on the height from which an observation is made.


Curvature of the Earth


An extension of thought, and generalization of one of the intermediate formulas in the steps described above, enables a practical determination of Earth’s curvature by posing the question: how much does the surface of the Earth depart from horizontal relative to a certain distance? See Figure 4.


Figure 4. Examining the curva- ture of the Earth by considering how far the Earth drops from a horizontal plane over a given distance.


we solve “h” [h = 0.0785 m/km2 x (1 km)2] and obtain 0.0785 meters, or 7.9 centimeters. The curvature of Earth is such that the surface drops nearly 8 centimeters from horizontal over a distance of 1 kilometer5. A critical point to note is that the relationship is not linear, and so the distance at two kilo- meters is not 16 centimeters, it is instead about 1.25 meters. And, at a distance of 8 km, the Earth drops off by 5 meters from horizontal.


Atmospheric Refraction


One final component necessary to discern a good approxi- mation of distance to objects on Earth is the refractive index of Earth’s atmosphere – or, how light is bent in the air in which we make our measurements and observations. The atmosphere is a fluid, and light bends as it passes through it (approximate index of refraction for air is 1.0003); in effect, this bending actually allows us to see farther than we could if there were no atmosphere. For example, the sun actually descends below the horizon before we see it go down, because light is bent upward and around the horizon allowing us to see it after it should have set. The exact atmospheric refraction depends on a number of factors, including temperature and pressure; however, a general correction can be estimated and applied to the curvature calculation to account for both atmo- spheric refraction and curvature6. The correction applied in surveying is typically between 1/6 and 1/7, effectively negating approximately 15% of the effects of Earth’s curvature (Brinker & Minnick, 1995) – meaning the perceived horizon is about 15% more distant than the geometric calculation determined earlier. A combined formula for curvature (C) and refraction (R) is therefore:


(0.15)(0.0785 m/(km2) + (0.0785 m/km2) d2 (0.011 m/(km2 ) + (0.0785 m/km2) d2 m/km2) d2


Mapping and Surveying Manipulate the equation below to solve for “h” now, instead


of “d” (again, as an approximation, we can use “rE” instead of “rE + h”).


2rE ) d2 2rE) d2/(2rE


Reducing the equation above, by dividing by two times the Earth’s radius, we arrive at:


d2/(2 (6371 (0.0000785 km(-1) d2 hm/km2) d2


We can now determine “h” in units of meter or kilometer, when “d” has units of kilometer. Suppose we wish to know the vertical drop from horizontal over a distance of 1 km: then,


Curvature and atmospheric refraction corrections are important to employ when determining elevation via trigono- metric techniques (e.g., trigonometric leveling); for example, during traditional surveying activities. Both contribute error to any attempt to determine elevation differences between points on the surface, and the deviations from horizontal are usually considered consequential at distances greater than about 500 feet.


A measurement of the difference in height between two locations is a customary exercise performed in mapping pro- cedures, such as in a geologic field techniques class or unit. As an example, consider Figure 5 of a right triangle, where angle phi (ø) can be determined and combined with a range finding for the distance from point “a” to point “b” to compute the distance “a” to “d” using trigonometry. The height of the plane table (H), or equivalent horizontal surface, is added to


5. To determine the value “h” in feet, use the conversion factor of 1 mile equals 1.61 kilometers to identify Earth’s radius in miles (3957


miles); then, by using h = d2/[2rE2 (when “d” is given in miles). To determine “h” in feet when “d” is given in feet, the equation becomes: h = 0.000126 /mi (x 1 mi/5280 222 with distance provided in thousands2 resulting in 0.666 feet per mile, and multiplying by 12 inches in a foot, the Earth’s surface drops from horizontal about 8 inches in a mile.


6. Interested readers are referred to entries for Young in the references for more details and specifics related to the physics and geometry of atmospheric refraction.


 


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