# Section Five: Geodetic Datums: Combining Reference Ellipsoids and Geoids

In Section Three, we looked at the idea that the Earth is not best represented by a sphere (or even a spheroid), but a geoid, which is a model of the variation in gravitational pull over the surface of the Earth. This variation allows for a modeling of true mean sea level at any single location on the Earth's surface, since an assumption could be made between the strength of the pull of gravity and where water would pool (higher gravitational pull, more water pooling).

We also noted the best way to simplify the shape is to match it to a 3D reference ellipsoid, one which either was best-fit for the whole world (global reference ellipsoid) or one which was best-fit for a local area (local reference ellipsoid - which could be regional, continental, state-wide, county wide, or city wide). We also learned that the reference ellipsoid was great for laying out the 2D portion of the map and we mentioned that the geoid is the base for the Z value or elevation portion for making maps. In order to make useful maps, we need combine the 2D, XY half of the map and the 3D, Z half of the map. This is accomplished with the creation of a geodetic datum, or really just "datum" for short.

### 2.5.1: Geodetic Datums

So far, we understand 2D maps, consisting of the shapes of continents, states, counties, or city limits (among a whole boatload of other features), drawn on the reference ellipsoid, while the elevation values are stored with the geoid. We also understand that the two are linked together to create a complete 3D picture of the area we are mapping. Now, we are going to look at how it is the two are linked to create geodetic datums, or - what most people say - just "datums".

Reference ellipsoids come in two varieties - global, containing a map of all of the continents and oceans, and local, which consist of any are smaller than the whole globe. Geoids are almost always, and in our studies here will always, consists of the entire globe. When we create datums, the reference ellipsoid is the half of the equation that decides the extent of the datum. If the reference ellipsoid only covers North America, it most likely fits inside the geoid pretty darn good in North America but doesn't touch anywhere else in the world (or if it does, by chance, it is ignored). When the datum is created, the extent - the boundaries of the datum - is limited to North America, regardless of the fact that the geoid maps the elevation of the entire planet. If the extent of the reference ellipsoid is the entire globe, it will fit mostly okay everywhere on Earth (the extent of the geoid) - some place will be very close, some will be quite a bit off, and some will be average. But that is nature of global datums - kinda good everywhere, but not really great anywhere.

As we will see in the next section, the reference ellipsoid is not blank to start, but actually covered with an evenly spaced Cartesian Coordinate System specifically called a geographic grid. This grid is used to start the process of labeling exact locations on the Earth's surface, but in this section, we will just remember the reference ellipsoid is not blank like a piece of printer paper, but covered in a grid like engineer's paper.

We haven't really talked much about the topographic surface - the surface upon which we walk around and observe the Earth's landforms, but we need to remember that all of these mathematical models are created and used with the goal of labeling and navigating to locations on the Earth's surface. Since that is our goal, we need something to connect the topographic surface to the geoid to the reference ellipsoid, and that real-world object is a benchmark. Benchmarks are, as we said, real world objects that are placed and maintained at specific and known locations of the Earth's surface. Some benchmarks are labeled monuments, some are chunks of rebar sunk into concrete, and some are just chunks of concrete, but all of them are scattered about world-wide and are used to connect the three surfaces together.

As, I'm sure you've noticed, there are not benchmarks covering the entire surface of the Earth, and most likely, you've never actually noticed a benchmark in real life ever. Since they are not covering the Earth at every location, we use the ones which do exist, then mathematically infer the rest.

Every benchmark is at a known location upon an existing Cartesian Coordinate System, we can use them as a connection point between the geoid (a model of the difference between GMSL and LMSL via a model of gravitational accelaration) and the reference ellipsoid (a gridded 3D shape with the oceans and land areas drawn upon it). If, for example, a benchmark existing on the topographic surface in the real world has an "address" (the exact known location) of -114.03, 34.42 which has been carefully surveyed and recorded, the geoid is created with GPS (an electronic means of finding Earth "addresses"), and the reference ellipsoid has a point marked on it's grid found by counting 34.42 intersections east of the 0,0 origin and 114.03 intersections south, we can simply connect the same location between the three different surfaces, creating beginning of a geodetic datum, or just datum. If we connect several benchmarks between the topographic surface with the matching, known coordinates located on a specific geoid and some specific reference ellipsoid, we have a pretty good start to the datum. If we know the "address" of each connected benchmark/geoid coordinates/reference ellipsoid's Cartesian Coordinate System coordinates, using simple counting we can label all the remaining intersections on the datum and call those mathematically dervied connections control points. And you thought you'd never use Cartesian Coordinate Systems again.....

• NOTE: Benchmarks are used for the basis of all datums. Once a benchmark is used to make the topographic surface-geoid-reference ellipsoid connection, it is "converted" into a control point. The difference between the two is a benchmark an actual object which resides in the real world and is used over and over to create lots of datums and a control point is a mathematically determined connection between a geoid and a reference ellipsoid which "resides" in the datum. So, within a datum, all benchmarks become control points, but not all control points started out as benchmarks.
FIGURE 2.17: An Example of a Benchmark
An example of a benchmark. While there is no single standard benchmark, they all include what they are, a height above sea level, and a cross or point in the middle for a surveyor to place the point of their survey equipment.
The Main Point...
Datums are: reference ellipsoids that have been linked to a specific geoid via control points, which connect real-world points called benchmarks with mathematically derived points on the reference ellipsoid.

### 2.5.2: Horizontal and Vertical Datums

We've explored the idea that the geodetic datum is the product of combining a specific geoid with a selected reference ellipsoid, starting with the benchmarks, and establishing all the remaining control points utilizing the principles of labeling intersections with a Cartesian Coordinate System. We noted that the reference ellipsoid - covered with a grid and with the 2D land and ocean masses drawn upon it - is connected to a geoid - a mathematical derivation of local mean sea level at any point on the Earth's surface via the specific gravitational pull at said point, which stores our 3D elevation values.

Looking at the creation of the datum in this way makes it seem like there are only "2D datums" and "3D geoids", when in reality, the two work together to produce three kinds of 3D datums: vertical datums,  horizontal datums, and three-dimensional datums. All three come from the initial product: combine a reference ellipsoid with a geoid via control points (both benchmarks and mathematically derived intersections) and each serves a specific purpose in Geospatial Sciences. All three are actually a 3 dimensional product, regardless of the fact that only one is called "three dimensional".  Everything we have looked at up to this point is really a 3D object.  Reference ellipsoids, datums, geoids ... all three dimensional objects.

Horizontal datums contain only XY values upon a 3D Earth, vertical datums contain only Z values upon a 3D Earth, and three-dimensional datums contain XY and Z values upon a 3D Earth.

Horizontal datums assume that every point on the Earth's surface is at an equal zero elevation - including the tops of mountains, the bottoms of oceans, and everything in between. Horizontal datums assume there is no change in topography or relief, and that the topographic surface is totally flat and level at every point on the Earth's surface.

Vertical datums come in three varieties: orthometric datums, which shows the changes in the Earth’s gravitational accelaration from 0 and any height referenced to the Earth’s gravity field can be called a geopotential height tidal datums, which show the changes in sea level due to tides and are based on local mean sea level (and remember, LMSL comes from the geoid); and three-dimensional datums, which combine horizontal datums with ellipsoidal height.

Three-dimensional datums are used by GPS units to determine a personal elevation at any given time. Introduction to GIS doesn't work with vertical or three-dimensional datums at all, and GPS for GIS only looks at the fact the if you stand at any given point on the Earth's surface, the elevation is known, so for now, just understand that there are a few products of the reference ellipsoid+geoid process.

### 2.5.3: Geodetic Height and Elevation

Elevation is defined as the measured distance between local mean sea level and the topographic surface. (as we've looked at while examining the geoid creation method, local mean sea level is specified because of the varied effects of the Earth’s gravitational pull; using a global mean sea level would create accurate measurements in some places and incorrect measurements in others.)

Altitude is defined as the height of an independent object, such as an aircraft, above ground level (AGL) or Above Sea Level (ASL). If an aircraft was flying at 2,000 feet AGL, it would be 2,000 feet above the ground right below it, no matter the elevation of that location.  Aircraft can also record values Above Sea Level (ASL), where elevation and altitude are combined.  Regardless of how an aircraft is recording measurements, it is not possible for a person to hike to an altitude, only an elevation.

Orthometric height is defined as measured distance between the geoid and the topographic surface.  Elevation measurements are created from and stored within vertical datums for accurate Z-values within a geographic coordinate system for accurate analysis.

Ellipsoid height is defined as the measured distance between the reference ellipsoid and the topographic surface.  GPS receivers use ellipsoidal height since the calculation is easier to obtain on the fly.  If the GPS knows where it is in relation to the XY positions on the reference ellipsoid, it can easily calculate it's height above that established zero.  For a GPS reciever, to calculate elevations based on the geoid is much more labor intensive.

Geoid separation (geoid height) is defined as the measured difference between the reference ellipsoid and the geoid. Worldwide, geoid separation varies from +278.87 feet (85 meters) to -351.05 feet (-107 meters), based on the WGS84 reference ellipsoid and the Earth Gravitational Model 1996 (EGM96) geoid. Geoid separation is used to increase the accuracy of GPS measurements in the post-processing phase of data collection since the GPS receiver uses elevation established between the reference ellipsoid and the topographic surface, but accurate measurements are between the geoid and the topographic surface (or the orthometric height).  In order to convert from ellipsoidal height to orthometric height, the geoid separation needs to be a known value.  Notice how these height determination definitions can be related back to the kinds of vertical datams.

Figure 2.18: A Graphical Explanation of Ellipsoid Height, Orthometric Height, and Geoid Height
Figure 2.19: Deviation of the Geoid and Reference Ellipsoid in For the World Geodetic Datum, 1984 (WGS84)
This image shows the areas where the reference ellipsoid and geoid deviate in the WGS84 Geographic Coordinate System.  We see that there are really only a few places where the two meet exactly; just the places where the pink and the blue meet in a single line.  Most areas of the geoid fall either above or below the surface of the reference ellipsoid, where the reference ellipsoid is a smooth, even shape and the geoid is the undulating weirdo shape.
The Main Point...
Representing the Earth in three ways, geoid, reference ellipsoid, and topographic surface, leads us to a need to know the measured distances between each of them. Each surface serves a different purpose in geodesy, so eliminating one is impossible.

### 2.5.4: Datum Shifts

Over time, data collection technology improves, while at the same time land and ocean masses are in constant flux - moving, shifting, and boogieing down. As a response to these changes, datum measurements regarding the precision of control points must also be changed to keep up with the shifting world. While it might seem rather obvious, it is a gazillion times easier to move the mathematical points on the datum then to go out, dig up, and physically replace the benchmarks. And since we know that benchmarks are used for several different datums, moving them to repair the accuracy of one datum would be catastrophic to all of the others.

datum shift is when the coordinate associated with a benchmark (and the resulting control points) is adjusted or changed based upon either better surveying techniques, better mathematical calculations, or adjustments for continental shift. Datum shifts can be major, noted with a two-digit year following the datum name — i.e. NAD27 to NAD83 — or minor, noted with a four-digit year in parenthesis following the datum name (i.e. WGS84(1988)). Major datum shifts are extremely involved, and usually include a major survey project and mathematical calculations, while minor datum shifts are usually completed when just a few control points are deemed incorrect.