Cosmic-Ray Navigation: Positioning With Muons Under Sea and Rock

Where positioning still fails

Satellite positioning is a radio service, and radio has two places it will not go: into seawater and through rock. GNSS signals sit in L-band, around 1.2–1.6 GHz, and conductive seawater swallows them within a few metres of the surface. A metre of granite does the same job. That is why a diver, a submersible, a tunnel crew and a cave rescue team all lose the one thing a walker takes for granted — an absolute fix that does not drift.

The existing answers each carry a bill. Acoustic baselines (LBL, USBL) work well, but someone must first deploy and survey transponders, and sound speed in water changes with temperature and salinity. Inertial navigation never loses lock, but it has no absolute truth in it: errors accumulate, so an INS is only as good as its last outside fix. Dead reckoning from a Doppler velocity log is bounded by the seabed being within range. All of them need infrastructure, calibration, or a recent fix from somewhere else.

Cosmic-ray muons are interesting precisely because they need none of that. They arrive everywhere on Earth, for free, from a source nobody controls — and they go through the water and the rock.

What a muon is, and why one reaches you underground

A proton from somewhere in the galaxy hits a nucleus in the upper atmosphere, roughly 15 km up. The collision makes pions; the pions decay into muons — heavy cousins of the electron, about 207 times the mass. At sea level you are standing in a steady rain of them: on the order of 60–70 muons per square metre per second per steradian above 1 GeV/c, which works out near 10,000 per square metre per minute. Several are passing through your body as you read this sentence.

Two properties make them useful. First, they are penetrating: the mean energy at the surface is only 3–4 GeV, but the tail of the spectrum punches through hundreds of metres of water and kilometres of rock — volcano imaging works out to about 5,000 metres of water-equivalent overburden. Second, they are fast and straight. A muon at depth travels at better than 0.9999999 c, close enough to the speed of light that range = c × Δt is a fair approximation, and it barely bends on the way: through 100 m of water the path deviates with a standard deviation of about 1.8 mm, and through 1,000 m about 4.2 mm.

There is a nice piece of physics hiding in that. A muon lives 2.2 microseconds at rest — long enough, at light speed, to cover only about 660 m. It reaches the ground from 15 km up because relativistic time dilation stretches its clock. Cosmic-ray navigation is, in a real sense, special relativity doing something practical.

muPS: trilateration on time of flight

The founding idea, published by Hiroyuki K. M. Tanaka (University of Tokyo) in Scientific Reports in 2020, inverts the satellite picture. Instead of transmitters in orbit, you place reference detectors at surveyed positions — on the surface, on a GNSS-fixed buoy, at the top of a shaft — and put a receiver detector wherever the position is unknown.

When a single muon crosses a reference detector and then the receiver, both record the crossing time. The gap gives the range directly: L = c × Δt. Collect ranges from three or more reference detectors and the receiver position (x, y, z) falls out of the same sphere-intersection algebra that GNSS uses — L_i² = (x_i − x_p)² + (y_i − y_p)² + (z_i − z_p)².

Everything then rides on the clocks. Light covers 30 cm in a nanosecond, so a nanosecond of timing error is 30 cm of position error, and the lab work is a catalogue of that fight: ±1 ns jitter in the bench experiment, and 50 ps to 700 ps once artificial cable delays of 50 ns to 5 μs were introduced. Even thermal expansion matters — 100 m of copper stretches about 1.7 mm per kelvin, worth roughly ±400 ps across a season.

With that under control, the first laboratory demonstration resolved a receiver to 24.5 mm in x, 45.8 mm in y and 11.1 mm in z. Centimetre accuracy needs about 1,000 muon events per fix — a number worth remembering, because it is the constraint that governs everything that follows.

MuWNS: cutting the cable

The original muPS synchronised its detectors over wire, which caps how far apart they can usefully sit. The muometric wireless navigation system replaced the cable with independent high-precision quartz clocks, and in June 2023 Tanaka's group walked it through a real building: four reference detectors on the sixth floor, a receiver carried through the basement corridors below.

It navigated — at 2 to 25 metres of accuracy, over ranges up to about 100 m depending on depth and walking speed. Tanaka's own assessment is the most useful sentence in the whole literature: “This is as good as, if not better than, single-point GPS positioning aboveground in urban areas… but it is still far from a practical level. People need one-metre accuracy.”

He named the fix, too: chip-scale atomic clocks. The physics does not need improving; the timekeeping does, and CSACs are still expensive enough to keep the technique in the lab.

MuWNS-V: use the direction, not just the clock

The 2024 follow-up (Varga & Tanaka, Scientific Reports 14:7605, 31 March 2024) took a different route around the clock problem. Instead of leaning entirely on time of flight, it uses tracking detectors that measure each muon's direction, then keeps only reference/receiver pairs whose tracks agree to within about 80 milliradians (4.6°). Random coincidences almost all fail that test, so the background collapses.

The result: 3.9 cm RMS with detectors 2.4 m apart, and 3.0 cm RMS at 3.3 m with a higher-resolution reference tracker — through 30–40 cm of concrete floor. Accuracy here is set by the reference tracker's angular resolution (6–15 mrad), not by the clock, which is why a plain temperature-compensated crystal oscillator rated 0.05 ppm is enough. A “cosmic time calibrator” keeps the two ends locked by spotting muons that cross both.

The honest caveat is the update rate: 0.3–0.6 Hz. That is one position fix every two or three seconds — fine for a slow robot or a fixed monument, useless for anything moving at speed. Muon rate, again, is the clock that actually limits you.

At sea: what has actually been measured

The strongest real-world evidence to date is not a navigation fix but a measurement that proves the coupling. The Tokyo-Bay Seafloor Hyper-Kilometric Submarine Deep Detector — muon sensor modules installed in the Aqua-Line tunnel, roughly 20 m below the seafloor under about 20 m of seawater — ran from 5 March 2021 and watched the muon count rise and fall with the tide, because a thicker water column absorbs more muons. Over 79 days its muographic sea level tracked the astronomical tide with a standard deviation of 12.85 cm, cleanly resolving 20–30 cm neap-tide swings. The array has since been used to spot meteotsunami waves.

That is the same physics a seafloor positioning system would exploit, run in reverse. The published proposal puts a 4.5 m diameter reference detector (16 m²) inside a GNSS buoy and the receiver inside a 27-tonne seafloor anchor, with three scenarios costed by detector area and patience:

• 100 m depth (volcanic monitoring): 1 m² detectors → a few centimetres within 100 days.
• 1,500 m depth (magma-driven deformation): 4 m² detectors → 10 cm within 100 days.
• 2,000 m depth (plate tectonics): 20 m² detectors → 5 cm within a year.

Read those numbers carefully: they are integration times, not fixes on a screen. At sea, muometric positioning is geodesy — watching the seafloor creep or a seamount swell without laying a single transponder — rather than a navigation display for a moving vessel.

Under mountains: the older half of the field

Using muons to see through rock is much older than using them to position. The first documented use was E. P. George in 1955, measuring the rock overburden above a tunnel by counting cosmic rays. In 1970 Luis Alvarez's team applied it to the Pyramid of Khafre at Giza and — famously — proved a negative, excluding hidden chambers in the region they surveyed. The technique came back into public view when ScanPyramids found a previously unknown corridor in Khufu's Great Pyramid, later confirmed by endoscope, and it has since mapped the bedrock under two major Swiss Alpine glaciers.

Volcanoes are where it gets sharp. A 2018 study of Sakurajima used a seven-layer multi-wire proportional chamber sited 2.6–2.8 km from the active craters, with 4 mm RMS position resolution and 2.7 mrad angular resolution. Over 157 days it logged 125.6 million events, resolved summit features 20–40 m across, and distinguished structures finer than 7.5 × 7.5 m² — imaging low-density regions consistent with magma pathways inside a mountain, from the outside.

Notice the symmetry with positioning. Muography holds the detector position known and solves for the rock. Muometric positioning holds the rock known — or irrelevant — and solves for the detector. One physics, two inverse problems, and the mountain that blocks your GPS is the same mountain that makes the picture.

The limits, stated plainly

It would be easy to write this technology up as a GPS replacement. It is not one, and the reasons are worth knowing:

• Flux is the clock. Roughly 1,000 events per centimetre-grade fix, and the rate falls about four orders of magnitude by 1,000 m of water-equivalent depth. Deeper means slower, always.
• Energy caps the depth. A 3–4 GeV average means the useful muons at great depth are a thin tail of the spectrum.
• Detectors are not pocket-sized. Gas chambers and scintillator stacks run from 30 × 30 cm to several square metres, and they need power and readout.
• Someone still has to survey the references. The system is only as absolute as the known positions you feed it — GNSS, in practice, at the surface.
• The models thin out off-vertical. Depth-intensity relations get unreliable past about 30° from vertical.
• Nothing here is real-time at speed. Sub-hertz update rates suit a crawling robot, a moored anchor, or a monument — not a vehicle.

The 2025 review of the field in Particles is blunt about the state of play: the concept dates from 2020, the follow-up literature is thin, and practical deployment is still an open question.

Where it plausibly lands

The realistic near-term role is not standalone navigation but the thing inertial systems have always lacked: an absolute fix that does not drift. An INS holds position beautifully between corrections and hopelessly without them; a muon fix every few seconds is exactly the correction a submersible or a tunnelling machine cannot otherwise get. The same logic puts it in mine and building collapses, where rescue robots have no map and no signal.

At sea, the geodesy case is stronger than the navigation case today — seafloor motion and seamount swelling, measured continuously without transponders, feeding tsunami and eruption warning. There are even proposals to use galactic cosmic rays for positioning, navigation and timing on the Moon and Mars, where there is no satellite constellation to fall back on, and interest in polar regions where GNSS geometry is poor.

The technology's one genuinely unmatched property is what it lacks: there is no transmitter. Nothing to jam, nothing to spoof, no constellation to lose. That is why navies and geodesists keep reading these papers even while the accuracy sits at metres.

Meanwhile, on the surface

None of this changes what you can do today. On the water and in the hills your position still comes from GNSS, and the accuracy you actually get depends on sky view, multipath and your receiver — you can measure it yourself with the GPS accuracy test, and how GPS accuracy works explains what the number means. If you are navigating at sea, GPS vs nautical chart covers the datum trap that puts a correct fix in the wrong place on paper. In the mountains, a grid reference is still what rescue asks for — grid references for mountain rescue.

Cosmic-ray positioning is a reminder of something the everyday tools hide: a coordinate is never free. It is always a measurement of something physical — a radio delay, an acoustic ping, or a subatomic particle that started its journey when a galactic proton hit the sky above you.