Barometric Altitude and Floor Signal-to-Noise

Ada re-derives this chapter’s own numbers step by step, at full precision

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ADA · CALCULATION AUDIT

Barometric Altitude and Floor Signal-to-Noise

The chapter’s altitude calculator defaults to a measured pressure of 900 hPa against a sea-level reference of 1013.25 hPa, feeding the barometric formula h = 44330 × (1 − (P/P0)^0.1903). It then argues that because each floor of a building causes about 0.36 hPa of pressure change and the BMP280 holds ±0.12 hPa relative accuracy, a climbing sensor should see a clean 3:1 signal-to-noise ratio per floor. This audit asks the question those numbers invite: does the exponential altitude formula agree with the chapter’s simpler 12 Pa/m rule of thumb, and does the 0.36 hPa floor signal really clear the sensor’s noise by 3:1?

Companion to the chapter Lab: Motion and Environment — every number here comes from that chapter.

See the relationship before changing it

The figure reads from left to right. The blue card is measured air pressure. The middle card applies this page's rule. The green card is barometric altitude. Walk the arrows once: set the input, apply the rule, then read the result with its unit.

The retained audit below checks several chapter fixtures. This model keeps those stated values fixed and changes only measured air pressure, so the numeric fixture does not switch without explanation.

Walk the arrows. Lower pressure raises inferred altitude along the atmosphere model rather than a fixed linear scale.

Derive the baseline in four named moves

1
Name the input. The chapter baseline is 900 hPa.
2
Name the relationship. height = 44,330 x (1 - (pressure / 1,013.25)^0.1903)
3
Substitute with units. 44,330 x (1 - (900 / 1,013.25)^0.1903) = 988.7 m
4
Read the result. Keep the unit beside the value. Use it only inside the technical boundary on this page.

Predict, then change measured air pressure

Try Predict the direction of height = 44,330 x (1 - (pressure / 1,013.25)^0.1903). Test another measured air pressure, then compare barometric altitude.

Observe Lower pressure raises inferred altitude along the atmosphere model rather than a fixed linear scale. Reset measured air pressure to 900 and compare barometric altitude.

Explain Lower pressure raises inferred altitude along the atmosphere model rather than a fixed linear scale.

Check yourself

What should you do before trusting a moved-control result?

Answer: Predict its direction, apply the shown relationship, keep the units, and reset to the worked baseline.

What does this small model leave out?

Answer: Only measured air pressure moves here. Field effects named in the technical boundary stay fixed.

Try

Treat 1013.25 hPa sea-level pressure and 900 hPa measured pressure as the fixed case; predict the altitude, then run Check audit.

Observe

The exponential result and 12 Pa/m shortcut separate by tens of metres, while a 0.36 hPa floor step reports exactly 3.0 against 0.12 hPa noise.

Explain

Air pressure changes nonlinearly with altitude, so a 12 Pa/m shortcut accumulates absolute-height error even when a local floor difference retains adequate signal-to-noise.

Ada: This chapter converts pressure to altitude two ways and then argues a floor is detectable. Both deserve full precision, because the barometric exponent hides how far the rule of thumb drifts. The formula is h = 44330 x (1 - (P/P0)^0.1903) with P0 = 1013.25 hPa.

Altitude at a measured P = 900 hPa:

Ratio: 900 / 1013.25 = 0.888231.
Raised to the exponent: 0.888231^0.1903 = 0.977697.
h = 44330 x (1 - 0.977697) = 44330 x 0.022303 = 988.7 m, which rounds to the chapter’s ~989 m.

Cross-check with the 12 Pa/m rule of thumb:

(1013.25 - 900) hPa x 100 Pa/hPa / 12 Pa/m = 11325 / 12 = 943.75 m, about 944 m.
The ~45 m gap is expected, not an error: 12 Pa/m holds near sea level, but at ~900 hPa the thinner air drops less pressure per meter, so the exponential result sits higher and is the trustworthy one.

Floor detection (each floor ~0.36 hPa, sensor relative accuracy 0.12 hPa):

Signal-to-noise 0.36 / 0.12 = 3.0, a clean 3:1 margin – and a 0.01 deg/s gyro bias would meanwhile drift 0.01 x 60 = 0.6 deg per minute, which is why orientation cannot lean on integration alone.

The audit shows the linear shortcut and the exponential formula disagree by tens of meters on purpose: a design that reports absolute altitude must use the exponential form, while floor detection needs only the 3:1 relative margin that survives the daily +/-30 hPa weather swing.

Every number above is taken from the chapter’s own material and re-derived step by step.

Technical boundaries: The standard-atmosphere conversion omits local weather calibration, temperature and humidity profiles, vertical airflow, sensor lag, building pressurisation, barometer drift, and correlated floor-to-floor disturbances.

Ready: work the ledger before checking it.