the figure of the earth

Brigadier Martin Hotine is not quite the image of a decorated officer. His name is styled with trailing acronyms that make it no surprise that there is an official portrait, yet in that painting he appears disheveled, his tie far off center in the collar of his jacket. He leans off to one side, not quite like he is sitting for a portrait, but more like he was caught in the middle of something. Photos of the man are often similar: he's distracted, looking down at his desk or staring into space. His mind seems to be elsewhere. Hotine had a lot to think about. His duties in the First and Second World Wars had only been a distraction from the real work of his career: the precise measurement of the whole British Empire.

Late in Hotine's career, he was honored not only by his own country (as a Commander of the Order of the British Empire) but by the United States as well (named an Officer of the Legion of Merit). Most of his awards, though, reflected the technical nature of his work: the Founder's Medal of the Royal Geographical Society, and shortly after his death in 1968, the Gold Medal of the United States Department of Commerce.

Ribbons and medals, though, do not quite capture the breadth of Hotine's work. His greatest memorial is an artifact of his work: squat concrete pillars surmounted by a triangular brass plate. Found atop mountains and hills throughout the United Kingdom, these "trig points," designed by Hotine himself, are the physical references of the Retriangulation of Great Britain. This effort, spanning from 1935 to 1962 with the interruption of WWII, revised an original triangulation (initiated in the 18th century) as the basis for British surveying. Through the course of this effort, Hotine developed methods that would revolutionize the field of geodesy. His collaboration with mapmakers from the United States, a continuation of his wartime surveying for the Allied Forces, set the stage for one of geodesy's most ambitious projects: a measurement taken across the Atlantic Ocean.

Geodesy is the field concerned with the measurement of the Earth. It is perhaps one of the greatest examples of the subtle complexity of the real world: superficially, the measurement of distances and areas is a simple problem. In practice, it is extremely complex, subject to a web of complications that mean that even the most modern efforts should be viewed only as close approximations.

To begin, we have to consider the shape of our planet. This question, "what shape is the Earth?," is a central topic in geodesy known as the "figure of the Earth," and it has occupied mathematicians, cartographers, and astronomers for centuries. Of course we know what shape the Earth is: it is a sphere. Well, that's true to a level of approximation, but one that isn't even close enough for highway construction.

Geodesy's foundations are in the measurement of angles and distances, taken from the Earth's surface—of course, for most of human history, where else would we take them? By measuring the angles between three points and performing some trigonometry, the relative positions of the three points can be determined. This is known as triangulation. The same is true if you measure the distance between three points, known as trilateration, but up until the development of electronics the measurement of very long distances was a far more difficult problem than the measurement of angles.

So, the first triangulation of Great Britain, conducted over some 60 years starting in 1791, measured the angles between mountain and hill peaks. These measurements were taken very precisely using a then-new instrument called a theodolite, which is essentially a telescope coupled to a protractor. By taking enough measurements between enough hilltops, surveyors formed a sort of mesh or web that slowly spread across the country. Eventually, this network of reference points was dense enough that locations of buildings, property claims, and enemy encampments could be stated accurately by their relative position to fixed reference points.

This explanation of geodetic triangulation has omitted a major problem. Three angles or three measurements can precisely define a triangle, but the solution depends on the surface over which the triangle is formed. The math is simplest in a flat plane. Over a sphere, it becomes more complicated but is still well understood. The Earth, though, is not a sphere. It's not even that close.

One of the reasons that it is difficult to define the shape of the Earth is that it is unclear exactly what shape you would refer to. The physical surface of the Earth, its topography, is extremely messy. There are mountains, there are valleys, and the whole thing is the result of long, stochastic processes that left behind something that is not amenable to a mathematical description. Besides, surveyors are often trying to establish where exactly the topography is (e.g. the altitude of a given point), so referencing those measurements against the topography itself would be tautological. The field of geodesy, as separate from mere measurement, is perhaps defined by this realization that measurements of the Earth must be taken in reference to an abstract plane.

The most tempting of these planes is sea level, more or less. Geodesists refer to this as the "geoid:" roughly speaking, the geoid is the shape that the Earth would take if it were covered entirely by water and undisturbed by tides, weather, geotectonics, etc. It is not a sphere because the Earth is rotating, and the centrifugal force pushes things outward at the equator, causing a slight "flattening" of the sphere into an ellipsoid. It is not an ellipsoid either, because the geoid is ultimately defined by gravity, and gravity is controlled by the density of the Earth nearby. For example, where the Earth's crust is very dense, gravity is locally greater and pulls the "water" of the geoid closer to the Earth's center. Where the Earth's crust is less dense, the geoid has a larger local radius. Ultimately, the geoid is irregular, describable only by measurement.

Instead, geodesists rely on a simplified form of the geoid called the ellipsoid. This is an ellipsoid that best approximates the geoid, for the surveyor's purposes. British mathematician and geodesist Alexander Ross Clarke, in the course of the 19th century Ordnance Survey, determined that an ellipsoid with a major axis of 6,378,306 meters and an inverse flattening ratio of 293.465 was the best approximation of the geoid based on the areas that the British Empire had surveyed so far.

Ellipsoids are not useful on their own, though, as they must be anchored to a location and orientation on the surface. In other words, an ellipsoid is just a shape, and you must have at least one reference point and angle, or otherwise multiple reference points, to define how the surface of the Earth is mapped to that shape. When you add a convention for how surface measurements are expressed (and relative to what, typically the same reference point at which the ellipsoid is defined), you have the basis of a geodetic system that can convert between abstract coordinates and physical locations in three dimensions.

One of the driving forces of cartography, both as an enabling factor and as a source of demand, is aviation. Aircraft cover large distances, and pilots can make close observations of the ground underneath them. As technology advanced, aerial photography offered a way to record the pilot's observations for later analysis.

Still, as with so many things in cartography, there is a chicken and egg problem. An aircraft can be used to survey a map, but the aircraft needs to know where it is—usually by comparison to a map. In a certain sense, a pilot setting out to map new territory is flying blind. Well, some pilots are literally flying blind, in fog or clouds or under the cover of night. Even before the war, aviation was venturing into more difficult missions and pilots found themselves caught in the dark.

A German company, Lorenz, developed the first solution for blind navigation: an early form of space-modulated radio beacon usually called a Lorenz beam. By emitting a precisely directional radio signal down the approach path to a runway, the Lorenz beam created an invisible path that pilots could follow. An instrument in the aircraft gave the pilot an indication of how far to the left or right they were, and when they were centered, they knew exactly where they were—at least on a line.

The first Lorenz beam was installed in 1932, and it opened the era of radionavigation. Other events in Germany would soon establish radionavigation as one of the period's most profound developments: a blessing to pilots and a curse to the people caught underneath.

This combination of an ellipsoid, reference points, and a system of measurement is roughly what geodesists call a datum. Clarke's ellipsoid, while most suitable in the UK where the majority of the measurements on which it was based were taken, was well-known in the very early 20th century when the United States Coast and Geodetic Survey set out to triangulate the United States. Almost 30 years of work led to revisions to the ellipsoid and the selection of a new reference point, Meades Ranch in Kansas. This became known as NAD27, the North American Datum of 1927, and remained the basis of most surveying in the US, Canada, and Mexico until a major revision adopted in 1983.

I explain this background to illustrate a core problem in geodesy: precise measurements require a datum, and a datum requires an ellipsoid, which is an approximation of the geoid calculated by fitting to a set of precise measurements. By the mid-20th century, major triangulations had been completed in the US, in the UK, and in many European countries.

The onset of the Second World War revealed a problem with this state of geodesy: the datums were all different. Military planners laid out maps surveyed by different countries and found that they did not meet up at the edges. Not just due to the projection, or due to the illustration, but because these mapping efforts fundamentally disagreed on the plane over which the maps were made. Different countries had used different ellipsoids (and different basic methods of determining the ellipsoids), they anchored them at different references, and they adopted different values for the correction of measurements. National surveys tended to be anchored somewhere in the interior, for centrality, which meant that datum-related inconsistencies were most acute at the borders of countries where different datums met.

An Army officer planning artillery fire would find that maps of the German-French border, produced by the mapping authorities of the two formerly separate countries, disagreed on the locations of towns by hundreds of meters. This was enough to become a dominant source of error in artillery aiming, so the military knew what it had to do: unify the maps of Europe onto a single datum. Soldiers on the front were joined by tactical surveyors, military units that traded rifles for theodolites as they completed one of the war's most important intelligence missions. This effort, to establish a uniform European datum, was not completed until after the war, when Allied intelligence units located the archives of the German cartographic authority and hauled them to the US for calculations.

The result became ED50, the European Datum of 1950.

The problem of flying blind was all the more acute for bombers. Early in the Second World War, Allied Forces had low expectations of German air defense and thought that bombers would be able to operate over Europe mostly unchallenged. Things did not work out so conveniently: German fighters were better than expected, ground-based defenses more extensive, and ultimately, bomber losses were much greater. This forced a change in strategy to night-time bombing runs. With the absent to primitive radar technology of the time, bombers were extremely difficult to intercept at night.

This sword cut both ways: bombing accuracy was also very poor at night as bombers struggled to find their targets, especially during defensive blackouts. German bombers headed towards England encountered the same conditions, forcing the same shift towards night-time bombing. Both sides put to work developing technology for improved night-time bombing. The Germans extended the Lorenz beam into a two-dimensional system that could "designate a target" by radio. On the Allied side, several systems were in simultaneous development in the UK and US. The rapid iteration of radionavigation technology and electronic countermeasures during the bombing of Great Britain became known as the "Battle of the Beams," and set the groundwork for much of our modern navigation technology.

Among the combatants in this ethereal battle was RCA, the Radio Corporation of America, which had unintentionally discovered a means of measuring distances by radio and then greatly improved it by incorporating features of a similar British system. RCA called it SHORAN, the Short Range Navigation system. SHORAN used an aircraft-mounted receiver that emitted pulses and measured the time until "reply" pulses arrived from fixed ground stations. By maintaining a constant delay time to a given ground station, an aircraft could fly a perfect orbit around it. By flying that orbit until a specified delay to a second ground station was achieved, an aircraft could position itself exactly on a bombing target. SHORAN saw extensive use in both WWII and the Korean War.

Let us consider the state of geodesy after the Second World War. By this time, the United States had been thoroughly surveyed under NAD27, the Retriangulation of Great Britain was underway using a datum called OSGB36 (Ordinance Survey of Great Britain, 1936), and European maps were being completed under ED50. Combined with new efforts in Japan, this put most of the Allied or Allied-occupied world into the same condition: accurate datums existed that covered large contiguous land areas, but the exact relationship between those land areas was difficult to determine.

In other words, the distance between two locations in the United States or Europe, even very far apart, could be fairly confidently stated because of the common datum. The distance between a location in the United States and a location in Europe, though, came up question marks. Cartographers, especially of the nautical charting variety, had methods of approximating distances over the ocean but they were of very poor accuracy compared to true triangulated surveys. Ocean charts were made mostly by celestial observation, a field that was struggling to develop into "astro-geodesy" because of the limited precision of telescope measurements. To put a scale on the problem, analysis of mid-century nautical charts against later methods found that many Pacific islands had mapped locations that were several kilometers off, and that even major land masses varied by hundreds of meters.

There is a simple reason for this problem: the methods of surveying. While experiments were conducted with intriguing methods like celestial photogrammetry, photographing the same balloon or flare against the starfield from multiple locations and calculating vectors to determine the distance between these locations, the only truly accurate method was still triangulation by theodolites. This was easy where survey stations could be set up on mountains and hills, but it was already a challenge in forested country where you simply couldn't see that far. It was impossible over the ocean, where there was no way of seeing from one landmass to another.

Up to the beginning of the Cold War, no one had paid that much effort towards surveying over the oceans. Geodesy was fundamentally a local concern: knowing where something was within its country was enough. The numerous countries of Europe and their extensive land borders had made the unified European datum a necessity, but even between Western Europe and Eastern Europe there was little need for standardization.

Geodesy did not become a global concern until the invention of the intercontinental ballistic missile.

By the end of the war, it was already obvious that SHORAN had potential beyond targeting bombers. SHORAN equipment had been tested off of the US coast, during which RCA engineers noted that a small Bahamian island must be several hundred yards away from its charted position. Later measurements with precision astronomical equipment confirmed the SHORAN-derived measurements as more accurate than the charts. A Coast and Geodetic Survey officer seconded to the Army Air Force for the war effort took note of this result, and reported back to the Survey as it was struggling to map the Aleutian Islands.

Many of the Aleutian Islands were close enough to be surveyed by conventional means, at least in theory. In practice, terrible weather and frequent fog meant that surveyors repeatedly tried and failed to spot one island from the next. The cost of flying people around, waiting for good weather, and getting a few measurements through each break in the fog had put the project well behind schedule. After the delivery of SHORAN equipment, a few months of experimentation developed a technique that completed the survey by radio.

At the same time, the oil industry was pushing into two frontiers: the South American jungle and the ocean offshore. Both were formidable challenges to surveyors, the jungle due to the lack of clear sightlines and the ocean due to the lack of anywhere to set up fixed points. Oil exploration companies purchased secondhand SHORAN equipment from the military and ran their own experiments, leading to refined solid-state radio equipment and another set of well-tested operating practices.

Early experiments in SHORAN surveying came up with a frustrating error: SHORAN measurements, as compared to reference points established by traditional surveying, were always a bit too short. It took careful comparisons of SHORAN networks to Coast and Geodetic Survey networks in Colorado and Florida to confirm the reason: the accepted value of the speed of light, up to that point, was incorrect by 16 ppm. In 1949, the same Coast and Geodetic Survey officer, Colonel Aslakson, published a new value for the speed of light derived from SHORAN. That measurement has held up to the modern day, within 1 ppm.

The British Ordnance Survey had maintained close contact with the Coast and Geodetic Survey, in part through Hotine, and eagerly observed these tests. The Retriangulation of Great Britain was back underway after the war, now with the benefit of improved technology, but it was facing similar challenges to those in the Aleutians. Some experimental use of SHORAN was attempted in Scotland. SHORAN surveying was widely adopted in Canada as well, where in the late 1940s Parliament authorized the first precise mapping of the entire country.

With all of these efforts in mind, and its own objectives as well, the Air Force continued research on SHORAN. Among other developments, this led to HIRAN, a high precision version of SHORAN that used improved receiver electronics to obtain much more accurate travel time measurements. HIRAN reduced the typical error of SHORAN distance measurements from hundreds of yards to around ten feet.

The Cold War was a fundamentally different type of conflict from the Second World War. Besides the somewhat speculative nature of the actual conflict, the distances involved were incredible: for the first time in military history, strategists planned out attacks that would start on one continent and end on another.

WWII experience had already shown the importance of accurate mapping for calculating SHORAN bombing targets, but this was less of an issue in the Cold War context since accurate geodesic data for the Soviet Union was a closely held secret and even the best of spies would struggle to operate a SHORAN ground station undetected. Bomber aircraft would have to navigate by celestial observations and then spot their targets more accurately by conventional means, which of course raised the same problems of night-time bombing. This was a contradiction in nuclear reprisal plans that was never really addressed, since by the time research was underway to achieve accurate global navigation the role of the bomber had been supplanted by the ICBM.

ICBMs used primarily inertial navigation, with corrections while underway from celestial measurements. Both inertial and celestial navigation were technologies that saw revolutionary advances due to the ICBM, reaching accuracies that were unimaginable during the Battle of the Beams.

There was a problem: the targeting capabilities of ICBMs were soon superior to the maps that were used to designate the targets.

In the era of the ICBM, the exact distance from, say, North Dakota to Moscow became a question of great significance. Military intelligence set about answering it, and a combination of espionage and information from the previous war allowed for the extension of ED50 into the Soviet Union with some degree of confidence, as well as the anchoring of Soviet maps to ED50. Even before this work was complete, though, it was obvious that with missile silos surveyed against NAD27 and targets surveyed against ED50, the actual misalignment between the two datums would become larger than the probable error of the missiles.

Post-facto, the Department of Defense presents a laundry list of motivations for a World Geodetic System. I'm sure there's some truth to all of it, we know now that accurate geodesy is useful in so many ways. In 1950, though, most of the applications hadn't yet been invented. Older materials tell a plainer story: the military needed to aim missiles, and to aim missiles it needed a single map that contained both the origin and the destination.

NAD27, OSGB36, and ED50 were all sufficient within their scope. What the Cold War demanded was a connection between the two. The Air Force determined that it would have to trilaterate the Atlantic Ocean.

Efforts started from both sides. Canadian survey efforts, already routinely using SHORAN, provided NAD27 references in the Maritimes. ED50 provided references in Norway, and the United Kingdom was well surveyed under OSGB36. If trilateration measurements could be made between Canada and the UK, and between the UK and Norway, that would be sufficient to relate all three datums to each other. This came to be known as the North Atlantic Tie.

To complete the tie, the Air Force would have to island-hop its way across: from Canada to Greenland, Greenland to Iceland, Iceland to Faroe Islands, then to Scotland and on to Norway. The first part of this project was completed between Scotland and Norway with the cooperation of the Ordnance Survey, but observations between Scotland and the Faroe Islands were the greatest challenge.

Here we will finally consider the details of SHORAN. SHORAN is a straightforward time of flight system in which ground stations support a transceiver on the aircraft. The aircraft emitted a pulse on some UHF frequency, and a ground station tuned to that frequency received it and immediately emitted another pulse. The aircraft equipment measured the time between transmitting the pulse and receiving the reply which was, much like radar, an indirect measurement of the distance between the aircraft and the ground station. By using two different ground stations on two different frequencies, and doing some math, it was of course possible to establish the position of the aircraft on a map.

More importantly to surveyors, though, it was realized that SHORAN could be used to accurately measure the distance between two ground stations. If you knew the altitude of the aircraft, and found the shortest possible SHORAN distance to the two stations, it was simple trigonometry to find the straight-line distance between stations. This method became known as "line-crossing:" surveyors would mark on a map the estimated straight-line between the two ground stations, and then draw a perpendicular line that crossed it in the middle. An aircraft would then fly back and forth on that line, taking repeated SHORAN distance measurements until the minimum sum was found. That point—where the total distance to the two stations was smallest—must be at the center point of the cross line. Those values were used, with the aircraft's altitude, to calculate the final measurement.

HIRAN, the enhanced accuracy of SHORAN, used the same basic principal but added a process called "gain riding" to zero out error due to the rise time of the transmitted pulses (which led to ambiguity in the "start" of the pulse). The details of gain riding have become obscure, but it was a manual process requiring a dedicated operator. I believe that the operator was simply adjusting the gain on the HIRAN receiving equipment to find the minimum point at which the pulse was detected at all—at which point the pulse must be tripping the detector at its maximum strength, and thus at the end of the rise time.

Combined with calculations to offset atmospheric effects and the electronic properties of the equipment, all of which had to be extensively researched during the refinement of SHORAN surveying, very fine accuracies could be achieved.

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Still, the realities of measuring long survey lines in the North Atlantic were a challenge. One veteran of an Air Force geodetic squadron tells of hours spent flying a modified B-50 Superfortress out of Florida to make line crossings along the North Atlantic Tie. The B-50 was used because it already had SHORAN equipment installed and plenty of room for the operators, but the distances involved in the island hops challenged the range of SHORAN and they had to fly as high as possible to improve their chances of receiving the reply pulses.

The B-50 was a heavy aircraft not designed for high altitudes, and so they thundered high over the Atlantic, engines at maximum power, losing altitude with every turn and then waiting to slowly regain it. A film camera, called a reconnaissance recorder, took repeated photographs of the SHORAN instrument's display as operators adjusted the receiver and logged weather and signal conditions for later use in making corrections. The aircraft was essentially war surplus, not in its top condition, and the heat went out.

You can imagine this group of unlucky airmen, flying back and forth over the ocean in an aircraft that was coming to match the outside air at 60 below Fahrenheit. With difficult reception conditions, atmospheric effects became more significant, leading to error. It took repeated crossings of every line to reach a target statistical confidence level in the result. Back and forth, colder and colder, freezing hands struggling to write out logs, until they got the okay to come back home. They had measured the distance between a ground station in Scotland, I believe in the outer Hebrides, and Iceland—over 550 miles. This was, at the time, the longest distance measurement ever taken by such direct means.

The North Atlantic Tie was completed in 1955. These are hard to judge, in part due to the intrinsic complexity of geodesy. Many measurements were taken of each line, multiple lines were measured between different ground stations, and multiple ground stations were set up on each island or continent. Some measurements were almost certainly accurate to within ten feet. On the other hand, the final trilateration network, after least-squares fitting, left a troubling large inconsistency in the distance between Norwegian reference stations that suggested some uncaught error.

The North Atlantic Tie was one part of a variety of data used for the computation of the World Geodetic System of 1960, or WGS60, the first global datum. WGS60 is a direct ancestor of WGS84, the datum used by the Global Positioning System and thus the de facto datum of global geodesy today.

We now know so much more of the figure of the Earth. WGS84 models the planet as an ellipsoid with semi-major and semi-minor axes of 6,378,137 and 6,356,752.314245 meters and an inverse flattening factor of 298.257223563. These three precise figures describe only the abstract reference plane, a utilitarian simplification of a geoid that we know to be so complex in its shape that the polynomial approximations used today, if fully expanded from the functions that generate them, would have millions of terms. These approximations are accurate enough to be their undoing; they become outdated as the shape of the planet physically changes from year to year.

WGS60 was the beginning of global geodesy, but it was also the end of global trilateration. By the time the computation of WGS60 was complete, satellite Vanguard 1 had reached orbit. Phase and Doppler measurements of its radio transponder could be used to indirectly determine the position of a ground station relative to the satellite, and then by extension to another ground station. Some of these measurements were incorporated into WGS60, and its fast-following revision WGS66 was heavily corrected based on satellite measurements. The space age had come, and with it, the era of long-distance surveying had gone.

The old school of geodesy had one last hurrah: WGS84 incorporated data prepared for NAD83, which heavily used the Transcontinental Traverse, a 1961-1976 effort that physically measured the United States by headings and distances. This was likely the last large-scale physical survey in the world. By the time it was completed, networks of satellite ground stations had been incorporated into various national geodetic systems, grounding the datums ironically in space.

Geodesy is a complicated field with many fascinating stories. For the sake of simplicity, I have left so many details out of this sketch of the first trilateration of the globe. Today, we take accurate geospatial information for granted, and military secrecy around geodesy is mostly a thing of the past. Our maps were hard-won, though, the result of decades of ingenuity and hard effort to answer questions that have troubled scientific thinkers for centuries. Magellan sailed around the globe, but he didn't know how far he'd gone. Hotine retired from the army to help found the Directorate of Overseas Surveys, for which he explored the world again... this time, counting every mile.