Chapter 2 Mapping Data

Written by Paul Pickell

You probably already accept that the Earth is “round” and not “flat”. You have probably held and touched a globe at some point in your life. But have you ever wondered how we describe location and measure something as large as the Earth? In this chapter, we will explore fundamental concepts for how we measure the Earth and orient ourselves with coordinate systems.

Learning Objectives

1. Understand the models of Earth’s figure and shape
2. Describe different vertical datums and how they are used to reference height
3. Understand the difference between cartesian, celestial, geographic, and projected coordinate systems
4. Recognize the differences among major types of map projections
5. Explore how projected coordinate systems distort and represent the world around us

Key Terms

Antipode, Great Circle, Small Circle, Geodesy, Vertical Datum, Horizontal Datum, Deflection of the Vertical, Ellipsoid, Spheroid, Geoid, Elevation, Orthometric Height, Geoid Height, Geodetic Height, Coordinate System, Celestial Coordinate System, Cartesian Coordinate System, Geographic Coordinate System, Projected Coordinate System, Map Projection, Tissot’s Indicatrix

2.1 Introduction to Geodesy

Geodesy is the fascinating science of measuring the shape, orientation, and gravity of Earth. Naturally, some of the questions that come to mind when thinking about such a grand topic are I thought the shape of Earth is a sphere? and How do we orient ourselves on Earth? and What does gravity have to do with mapping location?

All of these questions stem from need to represent location. For our purposes, location is the position of something relative to something else. In order to actually describe a location on Earth, we first need to know the size and shape of Earth. Some of the first estimations of Earth’s size and shape were made by Eratosthenes, a Greek mathematician from the second and third centuries B.C. Eratosthenes was responsible for many concepts we use in our everyday lives:

• Conceiving the first spherical model of Earth
• The first accurate measure of Earth’s circumference
• Calculating the tilt of Earth’s axis
• Calculating the distance of Earth to the Sun
• Invention of the leap day

Eratosthenes accurately calculated the circumference of Earth by noticing how the Sun shone directly down the bottom of a well in Syene (modern Egypt) at noon on the summer solstice. He later made a second observation at Alexandria at noon on the summer solstice with a pole and noticed a shadow. He measured the angle of the shadow and inferred the circumference of Earth, which was already known to be spherical (Figure 2.1).

Figure 2.1: Diagram showing how Eratosthenes estimated the circumference of Earth by observing the angle of a shadow that was cast about 800 km north of Syene in present-day Egypt. (Monniaux, cmglee, and jimhtatshawdotca 2005), CC-BY-SA-4.0.

Pretty simple, right? Turns out, Eratosthenes was off by only 75 km or less than 0.2% in his calculation! The actual North-South circumference of Earth is about 40,075 km. His calculation worked because the Sun’s rays are nearly parallel when they strike Earth. So if you observe the Sun at the same time in two locations on Earth on the North-South axis, you will notice the Sun has a different elevation above the horizon, which means different lengths of shadows will be cast on the ground. This is also a way to prove that the Earth is in fact round because a flat Earth would have equally-sized shadows everywhere at any given time of day.

2.2 Models of Earth

Here is a simple thought experiment to consider. Suppose you are trying to measure your own height. You probably have not given much thought about how to technically do this because it seems intuitive: place a measuring tape at the bottom of your feet and mark the measurement at the top of your head. If we break this down, there are some important rules to follow (Figure 2.2):

1. The measuring tape must originate somewhere. In other words, we need to define a reference point or surface of zero height (i.e., the ground).
2. The measuring tape must be a straight line and originate at a 90-degree angle, perpendicular to the ground.
3. The measurement must terminate at a point along an imaginary line that is tangential to your head, and yes, that line must be perpendicular to the measuring tape and also parallel with the ground.

Figure 2.2: Diagram for measuring height above a datum. Pickell, CC-BY-SA-4.0.

Whenever you measure your height, the ground is easy to define. It is whatever point you are standing on. This starting point it also known as a datum. A datum is simply a reference point, set of points, or a surface from which distances can be measured. It does not matter if you are below sea level, atop Mount Everest, or on the 30th floor of a skyscraper. You will always get an accurate and repeatable measure of your height using a datum that is defined directly below your feet. But what about measuring the height of terrain on Earth? Whenever we measure the height of Earth’s terrain above some reference surface, we are measuring elevation.

The same rules above apply when we measure elevation. In order for elevation measurements to be comparable across the world, we need to define a reference surface, a datum, for the entire planet. There are actually several ways that we can model the shape of Earth in order to produce a datum. Models of Earth’s shape are often referred to as either vertical datums (the plural of datum) if you are referencing elevation or horizontal datums if you are referencing location. A vertical datum is a 3D surface model that is used to reference heights or elevations for the Earth. A simple question like How high is Mount Logan in Yukon, Canada? is complicated by the need for a reference surface and the fact that Earth’s shape is irregular. In this section, we will review three types of vertical datums:

• Geodetic - based on geometry
• Tidal - based on sea level
• Gravimetric - based on gravity

2.2.1 Geodetic Vertical Datums

A geodetic vertical datum is one that describes the Earth’s shape in the simplest possible terms using standard geometry. Despite what a globe might lead you to believe, the Earth is not perfectly spherical, but it is close to being spherical. In fact, the radius of Earth varies by no more than 22 km or 0.35%, hardly anything you would ever notice if you were holding it in your hand. That small difference is, however, significant enough to lead to mapping inaccuracies at the local level if a spherical model of Earth was adopted (Figure 2.3). Instead, we frequently describe Earth’s shape as an oblate ellipsoid, which is essentially a sphere that has been flattened, and we define this ellipsoid with a semimajor and semiminor axis. Sometimes you will see the term spheroid used, which just means “sphere-like” and is interchangeable with the term ellipsoid.

Figure 2.3: Spherical geodetic datum. Pickell, CC-BY-SA-4.0.

There are many different ellipsoids that have been defined and are currently in use as datums. The most commonly used ellipsoid is called the World Geodetic System of 1984 or usually abbreviated as WGS 1984 or WGS 84. In fact, there are hundreds of ellipsoids that have been defined over recent centuries to model the shape of the Earth. The reason for so many other ellipsoids is due in part to technological advances that have improved the accuracy and precision of surveying as well as estimation of the ellipsoidal parameters. Many of these ellipsoids are not geocentric, that is, not originating from the center of mass of Earth. These datums are known as regional datums, which still describe the dimensions that approximate the shape of Earth, but are instead oriented so that the surface of the ellipsoid is congruent with a particular regional surface of Earth. For example, the European Datum 1950, the South American Datum 1969, the North American Datum 1983, and the Australian Geodetic Datum 1966 conform well to their respective continents, even better than WGS 1984 in most cases, but poorly anywhere else in the world.

Figure 2.4: Sphere versus ellispoid. Pickell, CC-BY-SA-4.0.

Figure 2.4 greatly exaggerates the flattening of the ellipsoid to illustrate the above points. In reality, the sphere is flattened using a flattening factor calculated as \(f=(CA-CG)/CA\) and defined exactly as \(f=298.257223560\) for WGS 1984. Thus, the semiminor axis (i.e., rotational axis) for the WGS 1984 ellipsoid (meters) is

\[ CG=CA-(CA×\frac{1}{f})=6378137-(6378137×\frac{1}{298.257223560})=6356752.3 \] where \(G\) is the North Pole and \(A\) is a point on the Equator. The sphere, of course, is much simpler where \(radius=CB=CA=6378137\).

2.2.2 Tidal Vertical Datums

A tidal vertical datum is likely one that you are familiar with. The premise of a tidal vertical datum is to use mean sea level as a reference surface, above which are positive elevations and below are negative elevations. This has a lot of advantages, like it is intuitive and oceans cover more than 70% of the planet’s surface so much of Earth’s land mass is near an ocean. However, the disadvantages are that sea level changes over time with tides and also with climate change. The not-so-obvious problem with a tidal vertical datum is that the sea level is actually not constant around the planet not only due to tides, but also temperature, air pressure, and gravity. In other words, mean sea level measured at a gauge station in Halifax on the Atlantic Ocean will not be the same distance from the center of Earth as mean sea level measured at Victoria on the Pacific Ocean (Figure 2.5). The primary challenge with a tidal vertical datum is extending it away from the coastline through a network of survey points using a process known as levelling, and even still, it is only meaningful during the epoch in which the mean sea level was measured at a number of tidal gauge stations.

Figure 2.5: Conceptual tidal datum for Canada. Pickell, CC-BY-SA-4.0.

2.2.3 Gravimetric Vertical Datums

The geoid is a physical approximation of the figure of Earth. The shape represents Earth’s surface with calmed oceans in the absence of other influences such as winds and tides. It is computed using gravity measurements of Earth’s surface and is best thought of as the surface or shape that the oceans would take under the influence of Earth’s gravity and rotation alone. In other words, the geoid represents the shape Earth would take if the oceans covered the entire planet. More specifically, the geoid is a gravimetric model of Earth’s shape that is defined as an equipotential surface from a constant gravity potential value. Due to the distribution of mass on Earth, gravity is not constant across the planet’s surface. As a result, the surface of Earth’s oceans is not smooth like a sphere, but instead undulates depending on where gravity forces water to remain at rest. You can think of Earth’s gravitational field as a series of parallel lines extending outwards from the center of mass of Earth into space. Any of these lines that you choose is an equipotential surface where the force of gravity is constant. Keep in mind that the force of gravity is stronger nearer the center of mass of Earth and weaker as you move away from it. Thus, the geoid is an arbitrary equipotential gravity surface that is chosen to roughly coincide with present-day mean sea level.

When you measure the height of something relative to a gravimetric vertical datum like the geoid, you must level your instrument. Levelling forms a vertical line that is orthogonal or perpendicular to the geoid, known as a plumb line. It is incredibly easy to visualize a plumb line. Simply tie a rock to the end of a string and hold the string with your outstretched arm. The length of the straightened string traces a plumb line to the center of mass of Earth, wherever you are. Because gravity changes with location on Earth and all plumb lines are converging on a singular point, plumb lines are never parallel. This phenomenon has important implications for comparing observations on the the ground with a geodetic model of Earth like an ellipsoid. In other words, the plumb line that you traced with your string is pointing to the center of mass of the geoid, but the center of the ellipsoid is often in a slightly different direction. This difference is known as the deflection of the vertical and is measured as the angular difference between the centre of the geoid and the centre of a reference ellipsoid. Like other measurements of geodetic location (i.e., latitude and longitude), the deflection of the vertical is comprised of two angles: \(ξ\) (xi) representing the north-south angular difference and \(η\) (eta) representing the east-west angular difference.

It should be evident by now that the reference surface that you choose as a vertical datum will determine the measured elevation of Earth’s terrain. Additionally, We frequently need to convert elevations between geodetic and gravimetric vertical datums. For example, when you use a Global Navigation Satellite System receiver, you are provided with an elevation that is relative to the WGS 1984 ellipsoid. The difference in height between an ellipsoid and the geoid is referred to as geoid height (N) while the difference in height between an ellipsoid and Earth’s surface is referred to as geodetic or ellipsoidal height (h). The difference in height between the geoid and the Earth’s surface is called orthometric height (H) (Figure 2.6), and is given as:

\[ H = h - N \]

Figure 2.6: Orthometric Height (H) is the ellipsoidal height (h) less the geoid height (N). Pickell, CC-BY-SA-4.0.

To illustrate the concept of a gravimetric datum, suppose we constructed a large, straight tunnel through the physical Earth that was tangential to the ellipsoid. If we allowed the oceans to flow freely through this tunnel, your experiences might convince you that water would flow from one end to the other. But in fact, this tunnel is so large, that the gravity field is changing. So the water would actually come to rest at the surface of the geoid or gravimetric model, as shown in Figure 2.7 below.

Figure 2.7: Thought experiment showing where water would be at rest within a tunnel through the geoid due to the equipotential force of gravity (g). Pickell, CC-BY-SA-4.0.

2.3 Case Study: The Canadian Geodetic Vertical Datum of 2013

The Canadian Geodetic Vertical Datum of 2013 (CGVD2013) is the current gravimetric vertical datum used in Canada to reference heights. It is defined with a potential gravity value of 62,636,856.0 \(m^2\) \(s^-2\). The pervious vertical datum in Canada - the Canadian Geodetic Vertical Datum of 1928 (CGVD28) - was actually a tidal vertical datum that corresponded to mean sea level measured at Yarmouth, Halifax, Pointe-au-Père, Vancouver and Prince-Rupert, and a height in Rouses Point in New York. It turns out that Halifax referenced to CGVD2013 is 64 centimeters below Halifax referenced to CGVD28!

For reference, CGVD2013 is 17 centimeters below mean sea level measured in Vancouver at the Pacific Ocean, 39 centimeters above mean sea level in Halifax at the Atlantic Ocean, and 36 centimeters above mean sea level in Tuktoyaktuk at the Arctic Ocean. The older CGDV28 did not have any survey benchmarks in the far north of Canada and, with the advent of more reliable satellite-based measurements, was modernized in 2015 to CGVD2013. The United States currently uses the North American Vertical Datum of 1988 (NAVD88), which was never adopted by Canada, but the United States will be modernizing their vertical datum by adopting a gravimetric model with the same gravity potential value as Canada as early as 2025.