Give me a minute to explain the second. In land surveying, positional accuracy will depend on how carefully you measure a second. If we move along a meridian on the surface of an ellipsoid, advancing from one latitude to another, then the distance moved is called a meridional arc (M). Let’s say that we are doing this in the area around Denver, at about 39.7 degrees North. And let’s set the distance traveled equal to one second of arc of latitude (from 39°42’00” to 39°42’01”). What distance would I have traveled on the ground in meters or feet?
 

Calculating the Size

First, we calculate the radius of curvature of the ellipsoid in that immediate area. It is given by:

ρ = a(1-e2)/(1-e2sin2Φ)3/2

(see Endnote)


For which we need the values for the semi-major axis a of the ellipsoid, and the ellipsoidal eccentricity e, the published size and shape of the GRS80 ellipsoid is given by:

a= 6,378,137.000 m, and

f = 1/298.2572222 = the flattening that we use to calculate eccentricity

e = 2f – f2 [1] Bomford, p. 564, eq. A.33

Therefore: e = 0.00669438


This we will use to calculate the above radius ρ:

ρ = 6378137 (1 – 0.006716862) /

(1 – 0.0067168622sin2(39°42’00”))

= 6378026.103 m


Almost done. We now approximate the meridional arc length between the two latitudes above as being simply:

M = ρ sin 1” = 6378026.103 * 0.004848137 = 31 m


Translating

If we express a position in terms of latitude and longitude only to the full second, such as 39°42’00”, then our positional resolution (latitude-wise) is only 31 meters (or 101 feet). Or, if we account for rounding up or down, the positional resolution is 15 meters (50 feet) at best.

If we want to express the position to the centimeter, we must increase the latitude resolution 31*100 times, or 3,100 times. This means that our latitude will have to have seconds expressed to an angular resolution of 1/3100” = 0.0003”, or four decimals, as in 39°42’00.0000”.

The same exercise can be done in the direction of longitudes, and the result is somewhat better since meridians converge toward the poles. Positional resolution for full-second readings vary from about 30 meters near the Equator to zero at the poles (about 24 meters at Denver).

How does this affect positional accuracy? That’s another story, one that deals with tools and methods. For more on good, practical geodesy, see https://www.geocounsel.com/services/training/.


Endnote:

[1] Bomford, G, 1971, “Geodesy,” Third Edition, Oxford at the Clarendon Press,  p.565, eq. A.53.