A seismometer measures ground displacement. From that signal, you can compute velocity (first derivative), acceleration (second derivative), and jerk (third derivative). Each step of differentiation amplifies high-frequency content and suppresses low-frequency trends. Each step should make the signal noisier, less informative, harder to interpret.
Journeau et al. (Nature Communications, 2025) monitored Piton de la Fournaise on La Réunion for a decade and found that the third derivative — jerk — is the most useful signal for predicting volcanic eruptions. Not displacement. Not velocity. Not acceleration. The rate of change of the rate of change of the rate of change.
The jerk signal detects very low frequency transients in horizontal ground motion and tilt, generated by fracture openings as magma forces pathways toward the surface. The amplitudes are a few nanometers per second cubed — invisible in the raw displacement record, buried under tectonic drift and ocean loading and thermal expansion. But each differentiation step strips one layer of slow-varying background. By the third derivative, the slow trends have been differentiated into insignificance, and what remains is the sharp, transient signature of rock cracking under pressure.
Over 24 eruptions between 2014 and 2023, automated jerk detection provided advance warning for 92%. Alarm times ranged from minutes to eight and a half hours before magma reached the surface. The false positive rate was 14% — and every false positive correctly identified real magma movement that simply didn't break through. The entire system runs on a single broadband seismometer.
The general principle: the most informative signal is not always the most direct measurement. Differentiation is usually described as noise amplification, and it is. But noise and signal have different spectral structures. If the signal is a sharp transient and the noise is a slow drift, differentiation is a filter that preferentially amplifies the signal. The third derivative works not despite being three steps removed from the raw measurement, but because of it. Each step strips background that the raw measurement treats as fundamental.
This suggests a broader diagnostic: when you can't find a signal in your data, the problem may not be that the signal is absent. The problem may be that the measurement frame — the variable you're plotting — buries it under structure that isn't noise but isn't relevant either. Taking the derivative doesn't add information. It changes which information is visible.