*Note: an edited and expanded version of this post is now online at ScientificAmerican.com.*

This video is no April’s fool joke: the Earth really is shaped like a potato. Only that the shape you see here is, hum, slightly exaggerated to highlight its irregularities. Another caveat: what is depicted here is not the shape of the planet, but rather the shape of an idealized sea-level surface extending around the entire globe — a surface that Earth scientists call the geoid.

The video is the most accurate reconstruction of the geoid to date, and was released by the European Space Agency at a scientific meeting in Munich on March 31 (see video of press conference here). It is based on data collected by ESA’s Gravity Field and Steady-State Ocean Circulation Explorer (GOCE). The five-meter long, arrowhead-shaped probe has been in a low orbit for almost exactly two years, taking painstakingly accurate measurements of the gravitational field of Earth.

Alas, many of the web sites and blogs that reported on GOCE’s results yesterday got it wrong. Contrary to what was stated in news stories and in blogs, the geoid is *not* a surface on which the strength of gravity is the same everywhere.

The geoid is, in the words of an oceanographer on the GOCE team, a surface such that if you placed a marble anywhere on it, it would stay there rather than rolling in any direction. Another way of saying it is, imagine you were an engineer traveling around the world with a spirit level. Then wherever you go, the level would be exactly parallel to the geoid at that place. Yet another equivalent definition: it is a surface that’s everywhere perpendicular to the direction of a plumb-bob, or in other words, to the gravitational field.

Gravity need not have the same strength everywhere on the geoid. In other words, if you could walk on the geoid you would see gravity always pointing exactly downwards, but your weight could slightly change from one region to another.

The misunderstanding may have stemmed from the confusion of two concepts from calculus: a function and its derivative. In this case, it’s actually two concepts from multivariable calculus: a vector field and its potential. The potential is the gravitational potential; the vector field is the gravity field. At any point in space, the gravity field represents the direction in which the gravitational potential rises fastest. Its magnitude, or length, is the rate of change of the potential. The field is called the gradient vector field of the potential and it is the 3-D version of a derivative. (To be pedantic: for historical reasons, the gravity field is defined to be the opposite vector to the gradient of the potential.)

If you were to follow a field line of the field – a curve which is at every point tangent to the vector field at that point – you would follow a curve of steepest ascent on the gravitational potential. That’s why it is called the gradient. (Note that because of inertia, the field lines of the gravitational field are not necessarily the trajectories of a body in free fall. The field won’t tell you in which direction you’ll be moving — only in which direction you will *accelerate*.)

Gradient vector fields are difficult to grasp in 3-D, because picturing the potential (to visualize the ascent or descent) would require a fourth dimension. But a 2-D analogy might help. Think of how water trickles down a hilly surface. Neglecting inertia, a droplet of water would follow a curve of steepest descent, which is a vector field made of vectors along the surface. In this analogy, the elevation of the droplet is the analogue of the potential at that point. And the analogue of the geoid is an elevation contour on a topographical map: the potential – the height — is constant along the contour. Obviously, some parts of this contour can be on steep terrain, whereas others can be on mild gradients. (On a topographical map, the slope is steeper where many different contour lines are crowded together.) The magnitude of the gradient vector field – which represents the steepness of a slope – is not the same at all points of the contour.

Thus, gravity does change in strength along the geoid, and so the geoid is not “a shape where the gravity is the same no matter where you stood on it,” as one blogger put it. Nor is it “how the Earth would look like if its shape were distorted to make gravity the same everywhere on its surface,” as reported by a major news site, which went on to add that “areas of strongest gravity are in yellow and weakest in blue,” apparently not noticing the contradiction with the previous sentence.

There are many reasons why the geoid isn’t a sphere. First and foremost, the Earth itself isn’t a sphere. It is closer to an ellipsoid, being flattened at the poles by the centrifugal force of its own rotation. But the planet isn’t an ellipsoid either, because of topography. Mountains and valleys are asymmetrical distributions of mass. That mass distribution affects the gravitational field and makes the geoid asymmetrical too.

Then there is the Earth’s inner and outer structure. The oceanic crust and the landmasses of the continental shelves are made of different materials that have different densities; and farther down, the mantle is not uniform either, with regions that have slightly different compositions and temperatures, and thus slightly different densities. “The variations,” Reiner Rummel, a senior GOCE scientist, told me in an email, “reflect processes in the deep earth mantle such as descending tectonic plates and hot mantle plumes.”

All of which affects the strength of the gravitational field, and the GOCE’s orbit. The probe measures these effects thus it measures the gravitational field.

The geoid is not to be confused with anomalies due to the topography of the land – although it is affected in part by it. Also – and here is where things get really tricky – it is not to be confused with the actual sea level.

Wait a minute, you say, but isn’t the ocean supposed to be at sea level by definition? Except for relatively small perturbations such as waves or tides, you say, shouldn’t the surface be exactly horizontal (that is, perpendicular to the gravitational field) everywhere? Isn’t that, after all, what “sea level” means? Yes, to some approximation. However, the oceans are not homogenous. Differences of salinity and temperature make them more or less dense. Moreover, the Earth’s rotation produces forces that keep the oceans in constant motion.

Just like the water in a river is not all at the same level – after all, it flows to the sea for a reason – the water in the ocean also has differences in height. And by height, it should be clear by now, I mean elevation relative to the geoid, which can be positive or negative. Oceanographers thus speak of the “topography” of the ocean, meaning that it really has hills and valleys. In fact, here is where the “ocean circulation” part of GOCE’s name comes into play: one of the main goals of the mission is to measure the oceans’ topography and from that deduce the structure of the oceanic currents.

The deviations of the geoid from the simplified, ellipsoidal model of the Earth are substantial: they range from 100 meters below (dark blue in the video) to 80 meters above (yellow), Rummel says.

The surface in the video is the geoid amplified by a factor 7,000, says GOCE mission manager Rune Floberghagen.