4 Locations in the Geospatial World
Explain the concept of a geodetic datum
Describe the shape of the Earth and how it affects spatial measurements
Identify common global and local datums
Recognize how differences in datums can lead to positional errors in GIS data
We live in a geospatial world, where we describe places using names, landmarks, or directions in daily lives. Across fields such as science, agriculture, business, urban planning, public health, civil engineering, and navigation, it is essential to go beyond simple descriptions and use precise, standardized systems to define and record locations on Earth surface. In such cases, we use coordinate systems to assign numerical values (such as latitude and longitude) to pinpoint every location across the globe in a systematic and consistent way.
A coordinate system provides a standardized framework for recording, analyzing and sharing spatial information. It enables us to accurately capture, analyze and visualize real-world processes, whether it is to examine the migration patterns of birds, map cropland performance, monitor real-time locations of airplanes, analyze the spread of an infectious disease, or identify hot spots in a city during a heat wave. Different agencies and organizations have developed various coordinate systems suitable for different tasks, such as global navigation or local mapping.
All spatial data—data with location information—is based on some form of coordinate system. To effectively use existing spatial data or create new datasets, we need to understand what coordinate systems are and how they are established. The first and essential step to develop such coordinate systems is to model the shape of the Earth. So, let’s start here.
4.1 Shape of the Earth
The development of modern space technologies makes it easier to study Geodesy—the fundamental properties of the Earth, including its geometric shape, its orientation in space, its gravity field, as well as the changes of these properties over time. For instance, the Earthrise Figure 4.1 was a photo taken by NASA astronaut William “Bill” Anders on December 24, 1968, during the Apollo 8 mission. For the first time, the photo showed the Earth—a beautiful, fragile blue sphere—rising from the barren lunar surface, revealing its shape, location and unity in the vastness of space. Fifty years to the day after taking the photo, Anders observed, “We set out to explore the moon and instead discovered the Earth.” However, people’s curiosity about the shape of the Earth goes far back in the history, long before these space technologies were available.
4.2 The rise of the spherical earth
Based on a few empirical observations, the ancient Greeks were able to establish a spherical earth. Aristotle (384 – 322 B.C.) provided some of the earliest empirical evidence for spherical earth, including 1) Ships disappear hull-first when they sail over the horizon; 2) Earth casts a round shadow on the moon during a lunar eclipse; 3) Different constellations are visible at different latitudes. Based on these observations, Aristotle concluded that “Again, our observations of the stars make it evident, not only that the Earth is circular, but also that it is a circle of no great size. For quite a small change of position to south or north causes a manifest alteration of the horizon” (Aristotle).
The ancient Greeks not only established the spherical earth but also calculated the earth’s circumference. In 200 BC, the Greek mathematician and geographer Eratosthenes noticed that objects in Syene (the city now known as Aswan located in Egypt) casted no shadows at noon in midsummer, indicating that the sun was directly overhead. Meanwhile, a pillar casted a shadow at the same time in Alexandria, which is located 489 miles north of Syene. The angle between the shadow and the pillar was 7.2 degrees. Based on the idea that the earth was a sphere, Eratosthenes reasoned the angle between Alexandria and Syene at the center of the Earth had to be 7.2 degrees, which is 1/50th of 360 degrees Figure 4.2. Based on this assumption, Eratosthenes calculated the circumference of the entire planet as
\[ \text{Circumference} = 50 \times 489 \text{ miles} = 24{,}450 \text{ miles} \]
This was very close to being right (the equatorial circumference is ~24,901 miles and the polar circumference is ~24,860 miles).
With the fall of Roman empire, knowledge about the shape and size of the Earth in Europe became fragmented. While Europe went through intellectual decline, the Islamic world was preserving and expanding upon Greek and Roman knowledge by translating and studying texts from these previous times (the Islamic Golden Age). Unlike figures such as Aristotle, who heavily relied on deductive logic, the Islamic scholars emphasized on empirical verification, believing that theories should match observations but not the other way around. They confirmed the spherical nature of the Earth and refined earlier calculations of Earth’s circumference. For instance, Persian scholar Al-Biruni made highly accurate calculations of Earth’s radius using trigonometry and the height of a mountain. His value of the Earth’s radius (~6340 km) was within 1% of the modern accepted mean radius of 6,371 km, or different of only 2% from the mean radius of curvature of the reference ellipsoid at the latitude of measurements (Sparavigna 2014).
4.3 The oblate or prolate spheroid?
It is not until the 17th century that development of physics, especially the understanding of motion and forces, drove new ideas about the shape of the Earth. René Descartes, the French mathematician and philosopher, proposed a prolate earth model, which is elongated at the poles and flattened at the equator. Based on different underlying physics theories, Isaac Newton and Christiaan Huygens argued for the oblate Earth model—a sphere flattened at the poles and bulging at the equator. In his 1687 masterpiece, Philosophiae Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), Newton calculated the radii of earth and reported that the earth is higher under the equator than at the poles, and that by an excess of about 17 miles. This value is close to the right number, which is 13 miles (Radius at the equator = 3963 miles or 6378 km; and Polar radius = 3950 miles or 6356 km). The debate between the two earth models was settled in the 18th century, when two competing French expeditions, one to the equator in Peru and another to the Arctic Circle in Lapland, were sent out to measure the length of a degree of latitude at different locations, which conclusively proved Newton’s oblate Earth model.
4.4 Geoid
The real resolution of the shape of the earth happened with the Space Age the 20th century. Satellite based earth observations provide a consistent method to measure the Earth’s shape and its subtle changes. Morden gravity missions such as the NASA Gravity Recovery and Climate Experiment (GRACE) mission can measure changes in the Earth’s gravity field, which is directly related to the distribution of mass on and within the planet, offering an invaluable method to measure the Earth’s shape. Using the Earth’s gravity field, we are now able to create a Geoid that is the constant gravity surface of the earth Figure 4.3.
The geoid is not a perfect, smooth sphere. Instead, it is a bumpy, irregular surface because the Earth’s mass is not distributed perfectly evenly. For example, areas with more mass (like large mountain ranges or denser rock underground) have a stronger gravitational pull, which causes the geoid to bulge outward. Conversely, areas with less mass (like deep ocean trenches or less dense rock) have a weaker pull, and the geoid dips inward.
The most fundamental property of the geoid is that it is an equipotential surface, meaning that at every point on the geoid, the force of gravity is the same, and it acts perpendicular to the surface. For a surveyor, this means a plumb line (a string with a weight at one end) will always be perfectly perpendicular to the geoid at any location, and a level surface (like a calm body of water) will always be perfectly parallel to the geoid. Thus, Geoid is often defined as a hypothetical Earth surface that represents the mean sea level in the absence of winds, currents, and most tides. It is what the ocean surface would take under the influence of gravity of earth only (not affected by any force other than Earth’s gravity, e.g., wind, waves, the Moon…). The geoid is the name given to the shape that the Earth would assume if it were all measured at mean sea level - close to but NOT equivalent to actual mean sea level.
4.5 Datum
While the geoid is able to capture the Earth’s shape in detail, it is too complex and irregular to model or use as the reference surface for locations on the Earth surface. Thus, an ellipsoid (aka spheroid)–a simplified smooth surface–is used instead . However, the discrepancy between the geoid and ellipsoid produces a source of error to locations.
Here, we define the term datum as a reference surface of the earth that is used for plotting locations across the globe. In this case, an ellipsoid with the origin in latitude and longitude and orientation is used to represent datum. The origin illustrates at which the ellipsoid is tied to a known (often monumented) location on or inside Earth (not necessarily at 0 latitude 0 longitude), indicating how to fit the ellipsoid to geoid. Orientation is how we position the ellipsoid, after we tied it to a known location – we still have some limited capacity to move it around relative to geoid.
Essentially, datum is a mathematical model to reference a coordinate system against Figure 4.4. In reality, people use different datums for different geographical areas, because the Earth has an imperfect, lumpy shape, and no single simple mathematical model can perfectly represent it everywhere. Thus, datums are designed to be the best possible fit for a specific region or for the entire globe. For instance, surveyors and cartographers would create a datum that provided the best fit for their particular country or continent to allow them to choose a reference ellipsoid that most closely matched the curvature of the Earth in that specific region. This approach made mapping and surveying highly accurate within that region, but it created significant problems when trying to connect different regional systems. For example, the North American Datum of 1927 (NAD 27) was a local datum that was extremely accurate for North America but would be offset by hundreds of meters or more if used in Europe or Asia.
In modern geodesy, it is also useful to distinguish between a datum and a reference frame. Here, a datum is the conceptual definition of the coordinate system (it specifies how a reference ellipsoid is positioned and oriented relative to the Earth), whereas a reference frame is its practical realization through coordinates assigned to measured control points, such as GNSS stations. In everyday GIS usage, however, the terms are often used somewhat loosely.
The common horizontal datums used in North America including the North American Datum of 1927 (NAD27), the North American Datum of 1983 (NAD83) and the World Geodetic System of 1984 (WGS84).
The NAD27 datum was developed for measurements of the United States and North America. It has its center positioned at Meades Ranch in Kansas. The NAD83 datum was developed by the National Geodetic Survey and Canadian agencies in 1986. It is used as the datum for much data for the United States and the North American continent as a whole. In the 1990s, the National Geodetic Survey (NGS), in collaboration with individual states, undertook a major project to update the NAD83 datum by establishing a new, more precise network of reference points across the United States (High Accuracy Regional Network of fitted points; HARN).
The WGS84 datum was developed by the U.S. Department of Defense and is used by the Global Positioning System (GPS) for locating points worldwide on Earth’s surface (a global datum). WGS84 has been revised six times since its original realization in 1987. The most recent version is WGS84 (G2139) implemented on 3 January 2021. Originally, the difference between WGS84 (as rolled out in 1987) and NAD83 (as first introduced in 1986) coordinates was so small that transformation was unnecessary. That is no longer the case when it comes to NAD83 (2011) and WGS84 (G1762).
In order to minimize coordinate changes over time, NAD 1983 is tied to the North American, Pacific (for Hawaii, Guam, etc.), and Marianas tectonic plates, while WGS 1984 is tied to the International Terrestrial Reference System (ITRF), which is independent of the tectonic plates. Over time, the two coordinate systems have become increasingly different and the difference can be ~1 to 2 meters within the continental United States.
Similar to the horizontal datum, a vertical datum refers to a surface of zero elevation to which heights (elevations) of various points are referenced using either an ellipsoid (ellipsoidal datum) or a specific geoid (orthometric datum) as a reference surface for height. Unlike a horizontal datum, a three-dimensional ellipsoidal datum provides the foundation for accurate determination of ellipsoidal height, which is the height of that same point of the Earth surface is the vertical distance from that point to the ellipsoid. On the contrary, orthometric height in North America is defined as the vertical distance measured from adopted geoid to the ground surface height, along a line that is always a right angle to all intervening equipotential surfaces. Considering the fact that the geoid is a surface of constant gravitational potential that closely approximates the mean sea level, whenever you see elevation data described as “X” feet/m above or below sea level, it is referring to the orthometric height. A related definition is the geoid height, which is defined as the difference between the geoid and ellipsoid, describing the height of the Earth’s geoid surface above or below an idealized reference ellipsoid Figure 4.5.
The North American Vertical Datum of 1988 (NAVD 88) is the primary vertical datum that has been used for elevations in the United States for several decades. It provides a common reference surface so that elevations measured in different locations can be compared consistently. NAVD 88 was established mainly through a large network of spirit-leveling measurements connecting thousands of permanent bencchmarks across the country.
4.6 National Spatial Reference System (NSRS) Modernization in the United States
While datums such as the NAD83 and NAVD88 provide the reference frameworks for horizontal and vertical positions, they both have limitations. For example, a major limitation of NAD83 is that its origin is not located exactly at Earth’s center of mass. It differs from a truly geocentric coordinate system by about 2.2 meters. This is important because modern Global Navigation Satellite Systems (GNSS), including GPS, use reference frames that are centered at Earth’s center of mass. Consequently, NAD 83 does not algin perfectly with global reference frames such as the International Terrestrial Reference System (ITRF). Regarding NAVD88, one important limitation is that it depends heavily on physical benchmarks (a.k.a. passive control points) that can be disturbed by construction, land subsidence, uplift, frost, or other processes, meaning that the published elevation of a benchmark may no longer represent its actual current position. In addition, NVAD 88 contains systematic errors from how the network was established. Because elevations were propagated across very long distance using leveling from a limited set of reference points, small measurement errors accumulated over space. Compared with a mordern gravity-based geoid, NAVD 88 has a known bias of roughly 50 cm and a broad tilt that can approach 1 m across the conterminous United States.
To overcome these limitations, in 2007, the National Geodetic Survey (NCS) announced plans to modernize the National Spatial Reference System (NSRS), which provides the coordinate framework used for surveying, mapping, and navigation in the United States. The NSRS modernization was designed to move away from a system that depends primarily on physical benchmarks and towards based on continuously operating GNSS stations, Earth-centered reference frames, precise gravity measurements, and modern geoid models. In this effort, the existing NAD83 and NAVD88 will be replaced using the modernized systems.
4.6.1 NATRF2022, PATRF2022, CATRF2022, MATRF2022
The NAD 83 datum will be replaced by four new terrestrial reference frames including NATRF2022 for the North American Plate, PATRF2022 for the Pacific Plate, CATRF2022 for the Caribbean Plate, and MATRF2022 for the Mariana Plate. These reference frames are defined relative to ITRF2020, making them geocentric and more directly comparable with GNSS measurements. Each frame rotates with its corresponding tectonic plate, which minimizes changes in coordinates caused by the large-scale motion of that plate.
4.6.2 North American–Pacific Geopotential Datum of 2022 (NAPGD2022)
The NAVD 88 vertical datum will be replaced by the North American–Pacific Geopotential Datum of 2022 (NAPGD2022 Figure 4.6) that will serve as the future geopotential datum of the NSRS. NAPGD2022 is based on measurements of the Earth’s gravity field and its primary reference surface is provided by GEOID2022, a high-resolution gravimetric model of the geoid. It encompasses three regions: the North American–Pacific region, Guam and the Commonwealth of the Northern Mariana Islands, and American Samoa. This datum provides the information needed to generate consistent orthometric heights, geoid undulations, gravity anomalies, deflections of the vertical, and other gravity-related geodetic quantities.
Together, these changes represent a fundamental modernization of the NSRS. The modernized system is designed to work more naturally with GNSS and to better represent the fact that the Earth’s surface is continually moving. Note that while the modernized NSRS is being rolled out and tested (2024-2026), the current NSRS, including NAD 83 and NAVD 88 etc, will remain the official NSRS of the United States.
4.7 Geographic Coordinate System (GCS)
Once the datum is fixed, we can use the datum to define a geographic coordinate system, which is a global reference system for determining the exact position of a point on Earth. In addition to a datum, a GCS includes a prime meridian that specifies the location of 0° longitude, and an angular unit often in degrees Figure 4.7. Latitude (a.k.a. parallels) is used to represent the east-to-west direction and equator is defined as the starting point of 0 degree latitude. Longitude (a.k.a. meridians) is used to represent the north-to-south direction (the North Pole to the South Pole). The International Meridian Conference (1884, US) defined the use of the meridian at the Royal Observatory in Greenwich, England as the zero reference line (prime median).
Since each datum defines a slightly different size, shape, and position of the Earth’s reference ellipsoid, a location’s coordinates (latitude and longitude) will also change when you switch datums. A datum shift describes the difference in coordinate values for the same point on Earth when measured against two different datums. This difference can range from a few centimeters to hundreds of meters, depending on the datums involved and the location of the point Figure 4.8. A datum transformation—a set of math formulas— can be used to convert point coordinates from one datum to another. However, errors up to several meters may occur during the transformation and errors accumulate with repeated transformations, because datum transformation is not an exact, one-to-one conversion but a mathematical process that uses a set of parameters (translations, rotations, and scale factors) to convert coordinates from one datum to another. These parameters are themselves estimates based on a best-fit model between the two datums, and they contain their own uncertainties. Thus, converting datums should be done only when necessary (use the fewest transformations possible), and care should be taken in choosing the best method. It is also suggested that using the same transformation each time to transform between these two datums. It is worth noting that dealing with some of the old datums may be problematic because their datum may not be well defined using the modern mathematical parameters.
Up to now, we have a system (GCS) that we can use to assign a unique set of coordinates for every location on Earth’s surface. This GCS can also be used for some other purposes. Longitude is closely related to time because the Earth rotates approximately \(360^{\circ}\) in 24 hours, or roughly \(15^{\circ}\) per hour and as the Earth rotates, different places face the Sun at different times. Here, we define solar noon as the time that the Sun reaches its highest point in the sky at a particular location. Because different longitudes face the Sun at different times, solar noon occurs at different clock times from place to place. Modern time zones are broadly organized around differences in longitude, with each time zone spanning 15 degree longitude. In practice, political boundaries and daylight-saving rules mean that time-zone boundaries do not follow longitude exactly Figure 4.9.
Because geographic coordinates describe locations on a curved Earth, the shortest distance between two points on the Earth surface is generally not a straight line as they do on a flat map. On a sphere, the shortest path follows a great circle, a circle whose center coincides with the center of the Earth (If you cut a sphere at one of its great circles, you would cut it exactly in half). The distance along this path is called the great-circle distance, which is the shortest distance between two points on the surface of a sphere. A great circle is also the largest possible circle that can be drawn around a sphere. Examples of great circles include the equator (the only parallel that is a great circle) and any circle formed by a meridian and its opposite meridian.
Great-circle distance is particularly important for long-distance navigation and global analysis. A route that follows a great circle may appear curved when displayed on some map projections even though it represents the shortest path over the Earth’s surface Figure 4.10. For more accurate GIS calculations, the Earth is usually treated as an ellipsoid rather than a perfect sphere. In this case, distances calculated on the ellipsoid are often referred to as geodesic distances.
4.8 References
Aristotle On the Heavens. The Works of Aristotle: Oxford University Press, 297-298.
Sparavigna AC. 2014. Al-Biruni and the Mathematical Geography. PHILICA.