Modeling Earth

From a young age, we are generally taught that Earth is a sphere. Images such as those taken from space (e.g., Figure 3.1.1) reinforce this idea. Yet, this is an oversimplification—Earth's actual shape is far more complicated as it appears to be. This poses an issue for cartographers, because our job often requires us to use measurements of the Earth to represent it accurately. So, short of going out and measuring every nook and cranny of the Earth’s surface, what tools can we use to create a faithful model of our very non-spherical planet?

Figure 3.1.1: The Earth from space.

Credit: NASA(link is external)

Due to the centrifugal force created by Earth’s rotation, Earth bulges outwards slightly at the middle—it is wider around the equator than from pole to pole. Because of this, a better way to describe Earth’s shape is as an ellipsoid. Ellipsoids which closely resemble spheres (also referred to as spheroids)—and as Earth is wider in the East-West direction, the most precise word to describe the approximation of Earth's shape is oblate ellipsoid (or oblate spheroid). In the literature about this topic, the terms ellipsoid and spheroid are often used interchangeably. Which term you use is less important than your understanding of the general concepts involved.

Figure 3.1.2: An example of an oblate spheroid.

Credit: USGS(link is external)

Many scientific fields are concerned with determining heights across Earth’s surface. In order to establish an accurate height, a zero-surface needs to be established. While the equator and prime meridian offer convenient zero-references for horizontal positions (a horizontal datum), heights (or the vertical component) is more challenging. To determine accurate heights (or elevation), a vertical datum is needed.

Elevation can be described as the distance of a point above a specified zero-surface of constant potential (usually gravity or gravitation). This distance is measured along the direction of gravity between the point in question on Earth’s surface and the specified zero-surface. To start this measurement, a suitable surface must be selected. Many surfaces exist such as an equipotential surface (i.e., a level surface of constant potential energy). Meyer (2010) explains that, in theory, on a level surface, there is no change in gravity potential and water does not flow across said surface. Water only flows between different equipotential surfaces due to forces that arise from the differences in potential energy.

Earth possesses an infinite number of equipotential surfaces. Deakin (1996) described that a cross-section of Earth’s equipotential surfaces would appear as an infinite number of thin onion skins that are not parallel to one another, are continuous, have smoothly varying radii of curvature, and are spaced closer together at the poles than at the equator with verticals as curved lines intersecting each surface at right angles. This convergence is a consequence of Earth’s physical oblate shape, to a first order and that gravity is stronger at the poles.

One particularly important zero-surface of the Earth is mean sea level (MSL), or the average height of the ocean’s surface. Historically, MSL was calculated simply by measuring the height of the ocean over time at fixed points. Mean sea level is affected by Earth’s gravity as gravity across Earth’s surface is not constant. Earth’s gravity is different in Lincoln, NE than it is in Los Angeles, CA, and if the ocean existed in both locations, its surface would be at different heights due, in part, to the influence of gravity. In places where the ocean is not found (like Lincoln, NE) it is difficult to measure MSL. Historically, estimates of MSL for interior locations were created by extending MSL based, in part, on long-term computations from tidal gauge stations located along the coasts. This process created a vertical datum called the Sea Level Datum of 1929 (which was later renamed to the National Geodetic Vertical Datum of 1929 in 1973). Unfortunately, the surveys used to create this datum introduced error into this geodetic height network.

Despite best intentions, MSL does not accurately represent Earth’s true shape. Estimates of MSL are fraught with accounting for forces such as winds, tides, currents that complicate the estimation process. A different zero-surface, known as the geoid, helps address these complications. In conceptual terms, the geoid is a level surface that the world’s ocean would assume if Earth’s rotation, winds, tides, and currents stopped, and its waters freely flowed over land conforming to Earth’s gravity field. In a general sense, MSL approximates the geoid, after a least squares adjustment is made. Van Sickle (2017) offers that these environmental forces cause MSL to deviate from the geoid up to 2 meters implying that MSL does not exactly follow the geoid. It is important to remember that the geoid does not represent the terrain of the Earth. However, the surfaces are related, for example, as rock masses have an effect on local gravitational forces and can help shape the geoid.

Figure 3.1.3 illustrates a geoid model. Overall, this geoid model is bumpy but these undulations are not visible to our eyes. In the figure, magenta hues indicate places of greater mass and stronger gravity (MSL is higher) while cyan hues represent lower mass and weaker gravity (MSL is lower). Some of Earth’s surface topography, reflected by the geoid undulations, can be visualized in this figure such as the Pyrenees Mountains and the Himalayans. Other undulations cannot and are attributed to, for example, different rock densities (such as observed in the basin in the Indian Ocean).

Figure 3.1.3: A graphic depiction of the geoid. This visualization is nicknamed “The Potsdam Gravity Potato.”

Credit: NASA(link is external)

Figure 3.1.4: The Geoid’s relationship with Earth’s surface and the reference Ellipsoid.

Credit: USGS(link is external)

The geoid is constantly changing—due both the ever-changing nature of Earth’s surface (e.g., from continental shifts, melting glaciers, etc.), and because technological advancements have allowed for more and more precise gravitational calculations over the years. The dynamics of Earth and the imprecision of measurement techniques mean that any model of the geoid is only an approximation (just as MSL is an approximation for Earth’s gravitational (and height) surface), in reality.