Section Three - Mathematically Measuring the Earth

Once historians measured the Earth, the next task was to create maps, used for navigation and exploration. In order to accomplish this task, they first needed to create a model of the Earth, upon which the oceans and continents could be represented, to scale and neatly labeled. The process in actually more complicated then presented in this class, but it is very important to understand the basics to understand how the GIS software accurately represents the Earth.

The process of modeling the Earth takes the topographic surface and simplifies it into a geoid (or a model of the variation between global and local mean sea level), which is then paired with a reference ellipsoid (where, as we will see, the actual map is drawn) to create a Geographic Coordinate System (or a means of creating a labeled grid for marking and navigating to locations on the Earth's surface, independent of language or local customs). Once we have a Geographic Coordinate System, we can make a whole series of accurate and precise maps using a projection method to create Projected Coordinate Systems. Both Geographic and Projected Coordinate Systems are a means of labeling locations and measuring land and ocean masses, just with different strengths and weaknesses.

2.3.1: Accuracy and Precision

Accuracy vs Precision
Accuracyis defined as the ability to hit your target

Precision is defined as the ability to repeat your results almost exactly the same every time

Error, in GIS and Cartography, is defined as the linear distance between the represented data and reality.

Results are considered both accurate and precise when the results are “on target”, every time. This concept is used in many scientific disciplines, but in GIS and Cartography, we use accurate and precise to define when our analysis and display are as close to the real world as possible, with as little error as possible.

2.3.2: Global Navigation Satellite System (GNSS)

The Global Navigation Satellite System or GNSS is a general term for the technology of using satellites and a signal receiver to pin-point a location anywhere on the surface of the Earth. Positional satellites orbiting the Earth are constantly emitting a signal, just waiting for a receiver to pick it up. People on Earth purchase GNSS capable units and use the signal to accomplish any number of location-based data collection, from creating a history of their latest run or mark out the boundary of a recent wildfire.

While GNSS is a general term, the system specific to lands under the jurisdiction of the United States is GPS[2] a term you are probably more familiar with. The United States owns and operates a current constellation of 31 satellites, launching a total of 68 different satellites since 1987. Like all thing mechanical, many of the 68 have been retired or failed (just two of those!) over the years. With a plan of constant coverage, plans for new satellites and future retirements are always in the works.

But how do the satellites “know” where you are? Trilateration[3]

You get all geared up for your run. You head outside with your favorite running app fired up and start to move. After a few minutes you look down and the darn thing still says “Acquiring GPS data”. You’ve learned from past experience that if you stop moving for a minute or two, the app will acquire that data faster then if you keep moving. Your phone is searching for not one, but at least three satellites to be emitting a signal in the area. As it finds one signal, the signal tells the phone the range of area which its owner could possibly be standing. When your smart phone finds a second signal, it calculates the possible overlap of the two maximum signal areas. Once a third satellite is found and the signal captured, your phone goes to work solving a triangle, noting the overlap of the three signal areas, much like a Venn diagram and where we get the name trilateration.

FIGURE 2.6: Venn Diagram

Your phone give you the green light and you’re off to the races. But it doesn’t stop there. For every additional satellite it captures, the possible area overlap shrinks and the pinpoint accuracy becomes even greater. If you kept moving while your smartphone was trying to capture satellite signals and trilaterate your location, the geometry of you location keeps changing, and while that is the whole point of GPS, to note your changing location and record the values, trying to solve the problem with a moving object is much harder.

FIGURE 2.7: Trilateration


2.3.3: The Topographic Surface

The surface of the Earth which you have been walking upon for your entire life is known in Geospatial Sciences as the Topographic Surface. This surface is the relief you see when you look out your window and is described using specific landforms. Relief can be defined as the difference between the highest and lowest point within a particular area while landforms are the descriptive words for individual features. Due to a lack of high mountains and deep lakes, parts of Kansas are often equated to pancakes where the Rocky Mountains have a great variation throughout, with high craggy peaks, deep lakes, and wide valleys.

As these are most likely concepts you already understand from, you know, being alive, these are the places and things cartographers and GIS technicians wish to map and measure. One person may need to know how far it is from The United Kingdom to the New World in order to pack supplies, while another needs to understand the area and topographic makeup of the Amazon Region to measure the loss or gain of rainforest over time.

Figure 2.8: Topographic Surface and Relief
A LiDAR Bare Earth Model: The surface of the the Earth with the trees taken away to show the topographic surface.Examples of Landforms

2.3.4: Geoids

In order to begin the process of mapping the Earth, cartographers and geodisists simplify the Earth just a bit using a mathematical model called a geoid. The idea of a geoid is most likely a new one to you as an introduction to GIS student, so let's first define it and then look at how one is created, and finish up with why it is used cartographers and geodisists.

Defining the Geoid

The geoid can be defined as: a model of the variation between global mean sea level (GMSL)and local mean sea level (LMSL) used to measure precise elevations on the topographic surface, or for now, a model of sea level to establish elevation values (a definition which we will expand on through this section).  You most likely have already interacted with elevation values in your daily life - it's everywhere - printed on city center signs, noted on the ground at airports, and used to measure all those 14'ers in Colorado (it's even where they get the name "fourteeners").  And you most likely also understand the concept of how elevation is measured: sea level is zero and the town or mountain is measured at some linear unit higher or lower, in the case of Death Valley, than sea level. But did you know that there really isn't an equal zero value all over the Earth's surface? Each time you are at the beach (or the shore, for you East Coast folks), you are indeed at 0 feet of elevation, but if you tied a string around the Earth, zero feet of elevation in Los Angeles is not at the same place as zero feet in New Jersey, meaning the local mean sea level is not equal. To understand the concept of a geoid, you need to first know the definitions of global and local mean sea level and understand just a little bit about how gravity and gravitational acceleration work, as what we determine as "sea level" is actually a reaction to the force of gravity and the measurement of object acceleration.

Newton's Law of Universal Gravitation states that any object with a measurable mass will pull any other object with a measurable mass towards it along a straight line. This means that the Earth is pulling you towards it's center along a straight path and you're pulling it right back, however, the mass of the Earth is much greater, the pull of the Earth has a greater effect upon you than you upon it. We call this force or the pulling action of the Earth against other objects of measurable mass, gravity and the measured result gravitational acceleration - simply, a measurement of the distance traveled by an object along a straight line over a known period of time, or looking at how long it will take an object to fall from a height above the Earth's surface to making contact with said surface. Newton's Second Law of Motion describes the relationship between force (the action) and acceleration (the effect of the force), stating they are directly proportional.  Historically, it was thought that gravity created a state of homeostasis, leveling out the oceans and making the sea level equal all over the Earth ... until it was figured out that gravitational acceleration is not equal everywhere on the Earth's surface. The average rate is indeed 9.81 m/s 2 - just like you may have learned in high school Physics, but since that is an average value, it must mean that there is a range of values which roughly centers on the learned value.  Gravitational acceleration at any given location on the Earth's surface varies based upon the materials and their densities which make up the inner and outer structures (themantel and crust, respectively) at that location - along with a few other things, but we are trying to keep it as simple as possible here, so we will just leave it as a factor based on the density of the material.

How are Geoids Measured?

To create a geoid (the variation between GMSL and LMSL), we need to first create a model of global mean sea level by removing all of the land masses and ocean tides and assuming the gravitational acceleration is equal everywhere on the Earth's surface.  This will result in a single, unmoving ocean which is perfectly flat and level everywhere.  From there, we need to measure the gravitational acceleration everywhere on the Earth's surface.  Traditionally, this was accomplished by taking several measurements with a gravimeter - a device which could measure the gravitational acceleration at a single point on the Earth's surface - and interpolating the remainder.  More recently, measurements with a higher accuracy using a less labor intensive method can be collected via  a satellite, as we will see in this YouTube video.

So, Why Use Geoids When it Comes to Maps?

Once we have all the measurements for every point covering the entire surface of the Earth and we have our model of GMSL, where the oceans are unmoving and level, we can either "push" down the model's surface to represent where the gravitational acceleration measurements are lower than the average or "pull" the water's surface up to represent those which are higher.  This pushing and pulling on the water's surface results in a Russet potato-like shape of the geoids.

Think about the pushing and pulling of the GMSL to create a geoid this way: if there is a finite amount of water and no land masses in our hypothetical perfect-sphere model of GMSL, water will be drawn towards the areas where the gravitational acceleration is higher (as the mass of the water is less than the mass of the Earth, so the Earth wins this Battle of Gravity).  If there is a finite amount of water, and that water is drawn towards areas where the gravitational acceleration is higher, this must mean the water must be drawn away from those areas where the gravitational acceleration was lower.  By moving the finite amount of water around on the GMSL model's surface, the resulting model is of local mean sea level, or the specific local zero position a measurement of elevation is based on, even inland where the ocean is not physically present.

It was stated at the very beginning of this chapter that the whole goal of maps is to create a graphical representation of our world to express the distribution of features (spatial data) and non-spatial data and that the first step in creating accurate maps is a means of labeling locations on the Earth's surface - creating a geographic coordinate system.  Since we live in a 3D world, it's important to have maps and systems that reflect this fact, and we express this factor via elevation measurements.  We will see in section 2.3.7 that maps are drawn upon the reference ellipsoid, but that only takes care of the horizontal mapping factor, the XY in our XYZ world. By using the geoid and attaching it to the reference ellipsoid, we are able to map all three measurement factors - X and Y to express where you are standing on the Earth's surface and Z to express the elevation at which you stand.

FIGURE 2.9: A Model of a Ellipsoid vs a Geoid


A model of the Average Mean Sea Level if the gravitational acceleration was equal everywhere on the Earth's surface (marked ellipsoid) vs the Geoid where the gravitational acceleration varies based on the density of the inner and outer Earth structures.

When we look at the model of the geoid in Figure 2.9, we see a full spectrum color ramp - a variation in color using all the colors of the rainbow starting at red and ending with violet - where the positive numbers are represented by shades red, zero is represented by green-blue, and the negative values are represented by shades of blue (kind of tough to see if you're using a black and white e-reader, but you can see an interactive color version of the geoid on the Learn GIS website). Areas where the model is red are areas where the gravitational acceleration is greater, thus the modeled variation in local mean sea level would be a value higher than the global average and areas modeled in blue are areas where the pull is lower and the local mean sea level would be less than the global average.

FIGURE 2.10: A model of a Geoid

Open in a new browser window to interact with a model of a geoid.  A great way to understand it better!

  • Click and drag the Earth to move it around
  • Zoom in and out with your mouse wheel
  • Adjust the height slider to change the visual exaggeration of the geoid
    • 0: no exaggeration, spheroid
    • 12: extreme exaggeration
  • Adjust the normal scale slider to change feature exaggeration
The Main Point...
Variations in the Earth structure density lead to a variation in the gravitational acceleration over the surface of the Earth. If there is a finite amount of water on the Earth's surface and water is drawn to where the gravitational acceleration is higher, we can then assume the local sea level at those areas is not equal to the global mean sea level; in fact it would be higher. Thus, a better model of where to place the zero elevation value at any given place on the Earth's surface should be based on the gravitational acceleration and the assumed local mean sea level and not the global average. This variable local mean sea level is modeled using a Geoid, where some places are modeled above the global average and in other places are modeled below. As our goal is to map a 3D world, having an accurate starting point for the Z value (elevation) is the first step to accurate and precise maps.

2.3.5: Ellipses and Circles

Geometry is the branch of mathematics which deals with the measurement of points, lines, angles, and surfaces, including defining both two dimensional and three dimensional shapes.  We use those definitions quite a bit (okay, literally all of the time) in geospatial sciences, and while geometry is not a prerequisite for a GIS course, there is a certain amount of expectation that a GIS technician understands some basic principles of how shapes are defined and expressed, as well as the understanding of basic shapes such as circles, squares, lines, points, and abstract shapes without any specific names.  Beyond that, the rest can be explained as we go.

At this point, there is an expectation that you understand that a circle is a two dimensional shape defined by a line which creates a closed loop where any point along the edge is an equal measurement from the center along a straight line (the radius) and that a sphere is the three dimensional version of a circle, where that radius measurement is equal for not just the XY plane, but also the Z.  The circle and the sphere is where we will start the discussion of ellipsoids and spheroids, the shapes we use to represent a simplified version of the Earth upon which we draw the geographic coordinate system.

An ellipse is a two dimensional shape (existing only in the X and Y planes, which are planes which are perpindicular to each other at a 90° angle) defined by a curved line forming a closed loop, where the sum of the distances from two points (foci) located on the major axis to every point on the line is constant. Okay, let's break that down so we better understand what in the heck was just said, but before that, let's define the parts of the ellipse so when we look at the breakdown of the ellipse definition, we know what we are looking at.  An ellipse is going to be longer in one direction than the other, meaning if you bisect the shape at the widest part, you've bisected it along the major axis.  If the ellispe is bisected along the shortest side, you've bisected it along the minor axis.  When each axis is bisected, they reach only from the center to the edge of the ellipse, resulting in the semi-major axis and semi-minor axis, respective to the major and minor axis.

FIGURE 2.11: Parts of the Ellipse


To examine the definition as a whole, bisect the ellipse along the major axis, and draw two focus points (as seen in the Figure 2.11 below - the major axis has points F1 and F2 marked).  Second, draw a single point anywhere along the edge of and draw a straight line from each foci to that single point (as seen with the orange point on the edge of the ellipse and red line a and blue line b).  Then we can observe two things: one - the length of red line plus blue line b are equal to the length of the major axis; and two - if you move the orange dot around the edge of the ellipse, as we see in the second and third images in Figure 2.11, the only thing that changes is the lengths of red line a and blue line b, the total length of a+b remains the same, which in turn means that the length of a+b still is equal to the length of the major axis.  Alright, there we go.  an ellipse is defined by a curved line forming a closed loop, where the sum of the distanced from two points (foci) located on the major axis to every point on the line is constant.

Figure 2.12: Ellipse
No matter where the orange dot is along the edge of an ellipse, the length of line a and line b are equal the the length of the major axis

By this definition, technically a circles are a just special case of ellipse where there are equal major and minor axis, meaning there is no major nor minor axis.  Instead, there exists a diameter and radius, representing the line which bisects the circle and measures from the center to the edge, respectively, as seen in Figure 2.3.A. The distance between two foci and any point along the edge of the circle still equal the diameter, regardless of where the point along the edge sits.  In Figure 2.13.B, the foci F1 and F2 each bisect the radius, meaning that lines a and b are each equal to the radius, and per the definition of a circle, twice the radius equals the diameter.

FIGURE 2.13: Circles are Just Special Case Ellipses

2.3.6: Spheres, Ellipsoids, and Ellipsoids of Revolution (Spheroids)

Two dimensional shapes become three dimensional when a measurement in the Z plane (the plane perpendicular to the X and Y planes) is added and in the case of ellipsoids, the relationship between the lengths of the axes along the X, Y, and Z planes defines the resulting shape.  In cases where all three axes are all different lengths, the resulting shape is an ellipsoid, while cases where two of the three axes are equal results in a ellipsoid of revolution, also known as spheroids, as seen in Figure 2.14 A.  Ellipsoids of revolution are called such due to the fact that two of the three axes remain invariant, or unchanging, when the object is  rotated along the third axis.  How the length of the third axis varies in comparison to the other two (which, remember, are equal to each other) is what gives each spheroid it's specific name.  As seen in Figure 2.14B, when the third axis is equal to the other two, the resulting shape is a sphere, when the third axis is shorter than the other two, the resulting shape is a oblate spheroid, and when the third axis is longer than the other two, the resulting shape is a prolate spheroid.   

FIGURE 2.14 A: A Model of a Sphere and an EllipsoidFIGURE 2.14 B: Ellipsoids of Revolution (Spheroids)
Ellipsoids are defined when two dimensional circles and ellipses are given a third, perpendicular measurement. When the measurements along the X, Y, and Z planes are unequal, the resulting shape is an ellipsoid and when two of the three axis are equal, the resulting shape is an ellipsoid of rotation, or a spheroid.Ellipsoids of revolution, AKA spheroids, are defined by how the third axis measures in relation to the two already equal axis.  When all three axis are equal, the result is a sphere; when the third axis is shorter than the other two, the result is a oblate spheroid; and when the third axis is longer than the other two, the result is a prolate spheroid.
The Main Point...
Using an ellipsoid over a spheroid when drawing a shape to best represent the planet takes into account the potato-like shape of the geoid, a necessary part of a geographic coordinate system.

2.3.7: The Earth as a Spheroid

Through years of history and ever-increasing accuracy of world measurements, the idea the Earth is a sphere was held as truth for thousands of years (Aristotle was recorded in 350 BCE stating the Earth must be a sphere based on the movement of constellations and we learned in the last section that Eratosthenes came up with his really close estimate around 200 BCE).  In 1687, Issac Newton's, whom we know was really into all things gravity, published Principia in which he stated the spinning mass of the Earth could not be a true sphere due to the fact the Earth was not completely solid, but closer viscous liquid on the inside, thus the true shape must instead be an ellipsoid of revolution, or a spheroid. While French mathematicians associated with the French Royal Academy of Sciences agreed with Newton that the Earth was indeed a spheroid and not a true sphere, their argument was the Earth must be a prolate spheroid (bulging at the poles, like a football) due to the physical property regarding the Conservation of Angular Momentum (the reason an ice skater is able to spin faster and faster with less effort the closer to their center they draw their arms). Newton argued based upon the semi-viscous core and the principle of centrifugal force (objects will fly away from a central point when force is placed upon them in a circular motion), the Earth must be a oblate spheroid(bugling at Equator, like a Frisbee).

The French Royal Academy of Sciences was eager to settle the score and set out two expeditions, one to the Arctic Circle and one to the Equator to measure the length of a degree of latitude (French Geodesic Mission, 1735) . The expedition and subsequent calculations determined the length of a degree of latitude at the Equator is 69.407 miles, while the length near the poles is 68.703 miles, proving the Earth is an oblate spheroid, slightly flattened at the poles an bulging at the equator.

The Main Point...
The Earth is better represented not with a sphere, but an oblate spheroid; the rotation of the planet plus the semi-viscous state of core result in a shape where the diameter along the equator is longer then the diameter between the poles.

2.3.8: Reference Ellipsoids

Geographic coordinate systems exclude the topographic surface in an effort to simplify the labeling process.  If we were to take the effort to create maps which perfectly mimicked the topographic surface, every time there was a change to the topographic surface, we would need to re-draw and re-calculate all of the completed systems.  As an answer to re-doing a ton of work after every earthquake or wind event in the Sahara Desert, geographic coordinate systems use the most simplified version of the Earth which still takes elevation into account.  Even though the Earth is indeed best represented by an oblate spheroid (which we know is a special case ellipsoid where the Z axis measurement is less than the measurements along the X and Y axes, which equal), our goal, instead, is to create a system which uses a geoid as a base measurement for the elevation.  Since the geoid is closer in shape to a Russet potato than an exercise ball with your little brother is sitting on it, it is a better idea to use an ellipsoid instead of a spheroid.  In the Section Four, we are going to start to build a geographic coordinate system, which starts with an ellipsoid and a geoid before combining them with other factors to create the entire system.

Yet, since this a compromise situation, using a smooth shape (an ellipsoid) to represent a lumpy shape (the geoid) will lead to situations where the two are not perfectly matched everywhere. To solve this problem, several reference ellipsoids exist in geodesy to best fit every portion of the Earth.  Global reference ellipsoids are designed to best-fit the whole world. These global reference ellipsoids have a general fit which fit pretty good everywhere and are not really a great fit anywhere. Other, smaller systems use a local reference ellipsoids which may be an excellent or almost excellent fit in one area but not even touch the geoid in another. Local reference ellipsoids can be considered local when applied to a region, continent, country, or state. Depending on the degree of accuracy and precision desired in regards to the fit, the proper reference ellipsoid must be chosen. The purpose of multiple reference ellipsoids is one of perfect fit based on the size of an area, trying to align as much of the geoid to the ellipsoid without variation as possible. As you can imagine, the smaller the ellipsoid, the better the fit to a single, focused area of the geoid.

FIGURE 2.15. An Example of Local and Global Reference Ellipsoids

NOTE: While we draw reference ellipsoids as what appears to be two dimensional shape, it must be kept in mind that reference ellipsoids are actually three-dimensional and fit inside the three-dimensional geoid.