The invisible lighthouse

A pilot in cloud over Pennsylvania in 1935 cannot see the ground, the Sun or the stars, and a ship feeling its way toward Nantucket in fog is no better off. Both can still hear. Every method in this chapter takes something you can measure in a radio signal, whether its direction, its travel time, a difference in travel times or a difference in phase, and turns it into a line, a circle or a curve on the chart. Where two curves cross, you are there.

That is the same logic as a pair of star sights in From a sight to a fix. What changes is the evidence. Celestial navigation measures an angle to a body whose position the almanac predicts. Radio navigation measures a property of a wave from a transmitter whose position is printed on the chart. The geometry of the answer depends entirely on which property you measure, and the chapter is organized around that idea: four measurements, four shapes of constraint.

Hear a direction§

The oldest radio aid needs no special transmitter at all. Any station with a known position will do: a coastal beacon, a broadcast mast, a lightship. What you need is an antenna that cares which way the wave arrives from, and a small loop of wire does exactly that.

Turn a loop until its plane points at the transmitter and the two vertical sides of the loop sit at slightly different distances along the wave, so their induced voltages differ and a current flows. Turn it broadside to the transmitter and both sides see the same field at the same instant; the voltages cancel and the signal disappears. The response follows a figure-eight:

where θ_arr is the bearing of the transmitter and θ_loop the direction the loop's plane faces. Operators used the null rather than the peak, because the peak is broad and flat while the null is sharp: near the null, the signal changes in proportion to the angle, so a few degrees of rotation take you from silence to a clear tone.

Try it. Notice two things. First, the null tells you the bearing of the transmitter only up to 180°: a loop is symmetric, so it cannot tell a station to the northeast from one to the southwest. On the chart the loop gives you a full line through the beacon, not a ray. Second, one line is not a fix. You could be anywhere along it. Mark a second station and the lines cross; how sharply they cross depends on the angle between them, and a crossing angle of 20° turns a 1° bearing error into a long, thin smear along the lines.

The ambiguity matters most when you have only one station and want to steer toward it. A small vertical antenna added to the loop's output changes the figure-eight into a heart-shaped cardioid with a single null. The cardioid's null is blunt, so in practice operators took the bearing on the sharp loop null and switched in the sense antenna only to decide which side the station was on.

Turning a large loop on a ship's mast was awkward. In 1907 Ettore Bellini and Alessandro Tosi filed a patent for a pair of fixed aerials crossed at right angles, coupled to two fixed coils, with a small third coil that could be turned between them^[1] documented. Their patent describes it as a way to aim a transmission without moving the aerials; used as a receiver, the same arrangement lets an operator find the null by turning the small coil instead of the antenna, a device called a radiogoniometer. Bellini–Tosi direction finders went to sea widely in the following decades^[2].

Fly the beam§

A direction finder makes the receiver do the work. The four-course radio range turned the problem around: the transmitter shaped its signal so that a pilot with an ordinary low-frequency receiver and a pair of headphones could tell whether the aircraft was on one of four fixed courses. The U.S. Centennial of Flight Commission's history calls it "the first practical radio navigation aid", introduced in 1928 and standardized by the Aeronautics Branch in 1929^[3] documented.

The station fed two crossed loop antennas. Each radiated a figure-eight like the one in Figure 1. One loop keyed the Morse letter A (· –), the other the letter N (– ·)^[4]. The two letters interlock: the gap in one is filled by the other. In a quadrant where the A loop is stronger, you hear A. Where N is stronger, you hear N. Along the four lines where the two figure-eights have equal strength, A and N merge into a steady, unbroken tone. Those four lines were the courses, and strung together across the country they became airways.

Notice how narrow the course is. Barry Schiff, who flew ranges, describes legs about 3° wide that became narrower near the station^[4]. The figure reproduces that by calling the tone steady when A and N are within about half a decibel of each other. Because the leg is an angle, its width in kilometers shrinks as you approach the station; tracking it gets harder exactly when it matters most. Directly over the mast, the signal fades into a cone of silence that told the pilot the station was below^[4].

Now add wind. Point the aircraft along a course and the tone holds for a while, then A or N creeps in as the wind drifts you sideways. The range tells you which side of the course you are on; it does not tell you by how much, how far you are from the station, or which of the four legs you are on. Pilots had a repertoire of orientation procedures to work out which quadrant they were in^[4], and misidentifying a leg could send an aircraft confidently along the wrong one. The Bureau of Air Commerce installed hundreds of ranges, and the last ones in the United States, in Alaska, lasted into the 1970s^[4] documented.

The same equisignal idea served landing. In Germany, the C. Lorenz company's approach beam laid two overlapping radiation lobes along the runway, one keyed with dots and the other with dashes, merging into a steady tone on the centerline; an experimental installation went in at Berlin-Tempelhof in 1932 and Deutsche Lufthansa adopted the system in 1934^[5] documented.

Radials and distance§

The range offered four courses. Its successor offered every course. Very high frequency omnidirectional range, VOR, was tested by the Civil Aeronautics Administration from 1944; by mid-1952 about 45,000 miles of VOR "Victor airways" were in place and the low-frequency ranges began to close^[3]. A VOR station transmits in the 108–118 MHz band, and an aircraft receiver compares the phase of two modulations to work out the direction from the station to the aircraft^[6]. That direction is a radial: a magnetic bearing measured from the station outward^[7].

Keep two words apart here. A radial is a property of where you are, a line of position. Your heading is where your nose points. You can sit on the 090 radial pointing north, south or anywhere else; the radial does not change until the aircraft moves. Confusing the two is the bearing-versus-heading mistake of Figure 1 in a new instrument.

Distance measuring equipment, DME, adds a range. The aircraft interrogates a ground transponder, the transponder replies after a fixed delay, and the aircraft times the round trip, in the 960–1215 MHz band^[6]. Subtract the known reply delay and the range follows from the speed of light, c = 299,792,458 m/s exactly^[8]:

The factor of 2 is the entire difference between radar-style ranging and the one-way timing of satellite navigation, and the one-way version needs a clock at the receiver that agrees with the transmitter's. The FAA began using DME across its whole system in 1961^[3]. A radial and a DME distance together give a fix from a single station: a line crossed by a circle. The Federal Radionavigation Plan of the early 1990s gave the predictable accuracy of VOR ground stations as 1.4° (2σ) and of DME as 0.1 NM^[6]. You can compare those two kinds of constraint in the radio laboratory.

Microseconds become geography§

Bearings degrade with distance: 1° of error is 1.7 km sideways at 100 km and 17 km at 1,000 km. Long-range navigation needed a measurement whose error does not grow that way, and timing provided it. Radio covers 300 m in a microsecond:

A pulse from a transmitter 600 km away arrives 2,001 µs after it leaves. If you knew exactly when it left, you would have your range from Equation 6.2. In 1940 no ship or aircraft carried a clock that could keep microseconds. The way around this was to stop asking how long the pulse took, and ask instead how much later one pulse arrived than another.

Drag P around. Notice that the absolute arrival times change as you move toward or away from both stations, but the gap between them only changes when you move closer to one than the other. A receiver needs no clock of its own to measure that gap; it needs only to compare two arrivals with a timebase stable over a few milliseconds. On wartime receivers that comparison was made by eye, lining up pulses on an oscilloscope.

The first system built on this idea was British. The Coast Guard's Loran-C handbook credits R. J. Dippy with proposing it in 1937; as Gee, it was in service in early 1942, with master and "slave" transmitters about 100 miles apart operating between 30 and 80 MHz^[9]. The RAF Museum dates its widespread introduction to Bomber Command from March 1942^[10] documented. Those very high frequencies propagate in nearly straight lines, which made the timing clean but limited Gee to roughly line-of-sight range, and the system was soon jammed over occupied Europe^[10].

In the United States, the MIT Radiation Laboratory developed the same principle at lower frequencies for longer range. The first permanent Loran pair, at Montauk Point, New York, and Fenwick Island, Delaware, was on the air by June 1942^[11]. The system later called Loran-A operated between 1,850 and 1,950 kHz, with ground-wave coverage of some 400 to 800 miles by day and sky-wave coverage as far as 1,400 NM at night; its fixes were good to about 1 NM on the ground wave and as poor as 6 NM on the sky wave^[9] documented. By the close of the war, the handbook counts some 70 transmitters and at least 75,000 receivers^[9].

Hyperbolas on the ocean§

What shape is the set of places where one pulse arrives a fixed time after another? If the two stations fire together, the time difference measures a difference of distances:

A curve on which the difference of distances to two fixed points stays constant is, by definition, a hyperbola, with the stations at its foci. On a flat map, every possible time difference gives one branch; the zero difference is the straight perpendicular bisector of the baseline, and the curves bend more sharply toward the nearer station as the difference grows. This is why the method is called hyperbolic navigation, and why Loran charts were overprinted with families of curves labeled in microseconds.

Here is what to notice in Figure 4. One time difference leaves you on a curve hundreds of kilometers long, exactly as one bearing left you on a line. A second station paired with the same master gives a second curve and a crossing. Sometimes two hyperbolas cross twice, and the second crossing is a legitimate solution that the navigator rejects because it is far from the dead-reckoning position; the figure marks it when it appears.

Now add noise. Each measured time difference is uncertain by some σ, so each curve becomes a band, and the fix becomes the small parallelogram where two bands overlap. Its size depends on two things. On the baseline between two stations, moving 150 m toward one changes the distance difference by 300 m and the time difference by 1 µs: the time difference is most sensitive to position there. Out beyond either end of the baseline, on the baseline extension, moving changes both distances by almost the same amount and the time difference hardly changes at all. A tiny timing error there implies a huge position error. The second factor is the crossing angle: two curves that meet at 90° pin a point; two that meet at 10° smear it along their common direction.

Both effects are captured by one number. Linearize Equation 6.4 at your position: a small displacement δP changes the range difference by g · δP, where g = û_A − û_B is the difference of the unit vectors pointing from each station toward you. Stack one such row per time difference into a matrix G, and the horizontal dilution of precision is

On the baseline extension g shrinks toward zero and HDOP grows without bound. The figure's geometry map is Equation 6.5 evaluated over the whole chart at low resolution, and the error ellipse is the same matrix read as a covariance: the least-squares solution of the two (or more) time differences carries the uncertainty (G^TWG)⁻¹, whose eigenvectors are the ellipse's axes.

Loran-C, the long-lived successor, was made operational in 1957 and placed under Coast Guard control in 1958^[9]. All its chains shared one frequency, 100 kHz, and were told apart by timing. A master transmits a group of nine pulses, eight spaced 1,000 µs apart and a ninth 2,000 µs later; each secondary then transmits a group of eight after a fixed emission delay, the baseline travel time plus a coding delay, so that the master's signal always arrives first everywhere in the coverage area. The interval between master groups, the group repetition interval, identifies the chain: 99,600 µs for the Northeast U.S. chain, designated 9960^[9] documented. Receivers did not only time the pulse envelopes; they also compared the phase of the 100 kHz carrier inside them, which is part of why Loran-C was so much more repeatable than Loran-A.

The Coast Guard handbook describes Loran-C as better than 0.25 NM in absolute accuracy within its coverage area^[9], and the Federal Radionavigation Plan put its repeatable accuracy, the ability to return to a position measured before, at 18 to 90 m^[6] documented. The gap between those two numbers is mostly propagation. A ground wave traveling over land is slower than one over sea water, so the time differences at a given spot differ slightly from the free-space prediction; charts and receivers applied additional secondary phase factor corrections, and a navigator who returned to a fishing spot by its recorded time differences could beat the charted accuracy. U.S. Loran-C transmissions ended on 8 February 2010^[12].

Phase is a ruler§

Pulses mark time with an edge. A continuous wave marks time more finely, with every cycle. If two transmitters radiate continuous signals locked in phase, a receiver can compare the phases on arrival; the phase difference is the range difference measured in wavelengths. The catch is that phase repeats. A receiver that sees one wave lagging the other by 0.37 of a cycle cannot tell 0.37 cycles from 1.37 or 12.37.

Each integer N gives a hyperbola, so the measurement φ puts you on one member of a whole family of them. The strip between successive in-phase curves is a lane, and on the baseline it is half a wavelength wide, because stepping x toward one station changes the distance difference by 2x.

This was the Decca Navigator. The American engineer W. J. O'Brien worked on the phase-comparison idea from 1936; it was developed in Britain during the war and first used operationally on D-Day, when the chain was switched on early on 5 June 1944 to guide the leaders of minesweeping flotillas toward the Normandy beaches^[13] documented. Walter Blanchard traces its origins in detail^[14]. The first civilian chain opened in south-east England in 1946^[13].

A Decca chain's stations all transmitted harmonics of one base frequency of about 14.17 kHz: the master at 6f (85 kHz on the English chain) and three slaves at 5f, 8f and 9f, called purple, red and green. A receiver multiplied each pair up to a common frequency before comparing them; the red pair was compared at 24f, giving lanes about 440 m wide on the baseline^[15]. The phase meters, called decometers, showed the fraction of a lane on a pointer and counted whole lanes as the ship crossed them.

Counting lanes works until it doesn't: a receiver switched off, an interruption at night or a burst of interference and the count is lost. Moving P in Figure 5 shows why the fine measurement alone is hopeless: dozens of lanes fit any reading. Decca's answer was lane identification, a coarser comparison on a much longer effective wavelength that tells you which lane you are in, provided its own error is less than half a lane. Lanes were grouped into zones of 10.6 km on the baseline, so the coarse system had to resolve one lane out of 24 on the red pattern^[15]. A partial lane-identification scheme arrived in 1947, and the "multipulse" technique of the mid-1950s removed the ambiguity almost completely^[13]. Raise the coarse error in the figure until it slips: the fix jumps by exactly one lane and looks as precise as before. That is the hazard of every ambiguous measurement, including the carrier-phase techniques in the next chapter. Decca's UK chains closed at midnight on 31 March 2000^[13].

The long-wave world§

Loran and Decca covered coasts and seas within reach of their chains. Omega tried to cover the planet with eight stations. It used very low frequencies, between 10.2 and 13.6 kHz, where signals travel for thousands of kilometers trapped between the ground and the ionosphere, and it was a phase-comparison system like Decca^[6]. At 10.2 kHz a lane was about 8 NM wide, and receivers combined several frequencies to widen the effective lanes^[6]. Omega was approved for development in 1968; six of its eight stations were operating by 1971^[16] documented.

Global coverage came at a cost. The height of the ionosphere changes between day and night, which changes the speed of the VLF signals and therefore the phase pattern; Omega users applied predicted propagation corrections. The Federal Radionavigation Plan gave its predictable accuracy as 2 to 4 NM (2 drms)^[6]: coarse by Loran standards, but available anywhere, which made it valuable for ships and aircraft crossing open oceans. Omega was shut down in September 1997^[16].

Terrestrial radio has not disappeared. Because satellite signals are weak and can be jammed, several countries have studied enhanced Loran, eLoran, as a backup. The United Kingdom and Ireland ran an eLoran prototype and trial service from 31 October 2014 until 31 December 2015, when the French and Norwegian transmitters it depended on went off the air^[17] documented. It is a useful reminder that a radio aid is an arrangement among governments and budgets as much as among transmitters.

Four shapes of a constraint§

Every system in this chapter answers the same question with a different measurement, and each measurement has its own shape on the chart. Figure 6 puts one receiver into four worlds at once.

The four families overlap more than the table suggests. Loran-C compared carrier phase inside its pulses. VOR measures a bearing by comparing phases. Every one of these systems leaves the same job to the navigator: combine at least two constraints, judge their geometry, and decide which crossing is real. You can try all four on one chart in the radio laboratory, where switching the measurement keeps the transmitters where you left them.

Two families are missing. One uses the Doppler shift of a moving transmitter, and its first great success was a satellite. The other times coded signals from transmitters whose clocks are so good, and whose positions are so well known, that the receiver can solve for its own clock error along with its position. Both are in You have four satellites.