Wave heights on the Cleggan – Inishbofin ferry

How this was measured

A phone held by hand, approximately level, top pointing aft, recording with strixie_mobile_service: accelerometer at 90 Hz, gyroscope at 52 Hz, magnetometer at 50 Hz, alongside a separate 1 Hz GNSS NMEA log.

1. Attitude. A Mahony complementary filter tracks the world "up" vector in the device frame — the gyroscope carries the fast, wave-band rotation and the accelerometer only trims slow drift, with a 50 s time constant. This matters: a plain low-passed "gravity" vector absorbs wave-band roll and pitch, and the residual tilt error injects g·θ of horizontal acceleration into the vertical channel — one degree of error is 0.17 m/s².
2. Vertical acceleration is the measured acceleration projected onto that up vector, minus its mean.
3. Heave by double integration in the frequency domain, with a −1/(2πf)² transfer function rolled off outside 0.07–0.50 Hz.
4. Statistics from the spectral moments (H_m0 = 4√m₀) and from zero-up-crossing analysis of the heave series.
5. Time alignment. The IMU clock is boot-relative and its wall-clock sidecar was not exported, so the offset was recovered by correlating tilt-compensated magnetometer heading against GNSS course over ground. Clear minimum, ±15 s ≈ ±85 m along track.

Why the numbers hold up

• The gravity estimate lands at 9.784 m/s² against 9.816 local — a 0.3 % scale error, irrelevant after high-passing.
• Band-limit insensitivity: H_m0 moves only between 2.25 and 2.52 m for any low cutoff in 0.05–0.09 Hz and high cutoff in 0.25–0.50 Hz, so the 1/f⁴ integration is not being driven by low-frequency noise.
• Rayleigh consistency: observed H_max/H_m0 = 1.62 against the expected √(ln N / 2) = 1.64 for 223 waves. The record has the statistics a real sea surface should have.

What it is not

• This is vessel heave, not the sea surface. For waves longer than the hull the heave response is close to 1:1; for the short end the ferry does not follow the water, so the record understates true wave height there.
• Periods are encounter periods, compressed by 11 knots into a roughly head sea. Correcting with ω_e = ω + ω²U/g puts T_z 5.2 s at about 7.7 s true and T_p 9.5 s at about 12.3 s — but that needs the wave direction, which one accelerometer cannot give. H_m0 is unaffected; the Doppler mapping preserves variance.
• Hand-held, not mounted. Arm motion is in the record, mostly above the wave band. What is plotted is in any case what a standing passenger felt.
• The barometer was useless here — flat wind-turbulence noise across the whole wave band, no peak. Dynamic pressure on an open deck at 11 knots swamps the 12 Pa/m hydrostatic signal. GNSS altitude too: 0.1 m quantisation and red multipath noise.

Files

• Static figure PNG
• Chart data JSON
• Summary statistics JSON
• Analysis source Python