Deflection of the Vertical
Source: GPS For Land Surveyors
How can latitude and longitude not define a unique position on the Earth? The reference lines—the mean equator and the zero meridian—are clearly defined. The units of degrees, minutes, seconds, and decimals of seconds, allow for the finest distinctions of measurement. Finally, the reference surface is the Earth itself. The answer to the question relies, in the first instance, on the fact that there are several categories of latitude and longitude, the geographical coordinates. From the various options, astronomic, geocentric and geodetic. Their definition has a good deal to do with where down is.
To begin that discussion of geodetic reference frames (datums), we'll talk about the deflection of the vertical. Please notice the astronomical coordinates of a station in the illustration. This is an actual station called Youghall in North America. The astronomic coordinates of that station, 40 degrees, 25 minutes, 36.28 seconds, differ from the geodetic coordinate, the latitude in that case being 40 degrees, 25 minutes, 33.39 seconds and the longitude in astronomic, 108 degrees, 46 minutes, 00.08 seconds, and the longitude in geodetic coordinate, 108 degrees, 45 minutes, 57.78 seconds. These coordinates differ because down is governed by gravity when it comes to astronomic coordinates. However, down is perpendicular to the ellipsoid for geodetic coordinates.
The ellipsoid is something that we're going to have to discuss. In shorthand terms, it is a smooth, mathematically determined representation of the Earth. It isn't plagued with the irregularities of the actual terrestrial surface, nor the irregularities of gravity. Please do not think that latitude and longitude coordinates are independent of the reference frame (datum) or independent of the ellipsoid that you are discussing.
Source: GPS For Land Surveyors
Here is an illustration, in the upper part we have an instrument that's set up on the Earth's topographic surface, and the red dashed line is perpendicular to the ellipsoid at its point, at the point where we find the instrument. The black line is the direction of down according to a plumb line (gravity), and the deflection of the vertical are the angles between them. However, the geodetic latitude, the version most used, is based upon down according to perpendicular to the ellipsoid, the red dashed line. Then, there's a third kind of latitude, the geocentric latitude. This is indicated by the line from the point on the Earth's surface to the geocenter, or the center of mass of the Earth (the green dashed line through the geocenter).
Geocentric, Geodetic, and Astronomic Latitude
Each can be extended upward, too, toward the zenith, and there are small angles between them. As mentioned, the difference between the vertical extension of a plumb line and the vertical extension of a line perpendicular to the ellipsoid is called the deflection of the vertical. It sounds better than the difference in down. Even though the discussion has so far been limited to latitude, the deflection of the vertical usually has both a north-south and an east–west component, so the deflection of the vertical has an effect on azimuths. There will be a slight difference between the azimuth of a GPS/GNSS baseline and the astronomically determined azimuth of the same line.
Geodetic and Geocentric Latitudes
Source: GPS For Land Surveyors
In one way of looking at it, GPS/GNSS is the reason that this issue has arisen so prominently in our work. Before GPS/GNSS existed, determination of latitude and longitude was done primarily with astronomic observations, which were done with optical instruments that were set up on the surface of the Earth. Imagine an optical surveying instrument set up over a point. If it is centered precisely with a plumb bob and leveled carefully, the plumb line and the line of the level telescope of the instrument are perpendicular to each other. In other words, the level line, the horizon of the instrument, is perpendicular to gravity. That optical instrument set up so carefully cannot be used to measure geodetic latitude and longitude directly, because they are not relative to the actual Earth. They are realtive to an ellipsoidal model of the Earth, and it is quite impossible to set up an instrument on that model. Furthermore, gravity does not even come into down on the model as it is perpendicular to the ellipsoidal surface at a particular point, whereas down is the direction of gravity at the point on the real Earth. They are most often not the same thing. Fortunately, the difference is usually small.
Nevertheless, using an instrument so carefully oriented to gravity, it is possible to determine the astronomic latitude and longitude of the point. Measuring the altitude of a circumpolar star is one good method of finding the latitude of the point from which the measurement is made. The measured altitude would be relative to the horizontal level line of the instrument, of course. One might expect that this astronomic latitude would be the same as the geocentric latitude of the point, but they are different. The difference is due to the fact that a plumb line coincides does not point to the center of the Earth where the line used to derive geocentric latitude originates. So there is a very long history indeed of geographical coordinates being derived from astronomic observations. Yet, the most commonly used coordinates are not astronomic latitudes and longitudes, but geodetic latitudes and longitudes. So, conversion to geodetic coordinates has a long history, too. Until the advent of GPS/GNSS, geodetic latitudes and longitudes were often values ultimately derived from astronomic observations by post-observation calculation. For example, the National Geodetic Survey and others corrected those coordinates and made them geodetic. In a sense conversion is still needed but GPS/GNSS receiver can display the geodetic latitude and longitude of a point to the user immediately because the calculations can be completed with incredible speed.
The fundamental fact remains unchanged: the instruments by which latitudes and longitudes are measured are oriented to gravity; the ellipsoidal model on which geodetic latitudes and longitudes are determined is not. And that is just as true for the antenna of a GPS/GNSS receiver, an optical surveying instrument, a camera in an airplane taking aerial photography, or even the GPS/GNSS satellites themselves. The most important point to carry away is this: latitudes and longitudes are dependent upon the reference frame (datum) on which they're measured. The secondary point is that, when we speak of latitude and longitude, we almost always mean geodetic latitude and longitude based upon an ellipsoidal model of the Earth.