Volcano Failure Forecast

What is modelled

Before some eruptions the rate of small volcano-tectonic earthquakes accelerates. The failure forecast method treats it like a material approaching failure: the rate obeys dΩ/dt = A Ωα, so it diverges at a finite time. For α = 2 the inverse rate falls on a straight line, and the day that line reaches zero is the forecast of the eruption.

Equations

• Voight law dr/dt = A rα, solved by r(t) = r0 (1 − t/tf)−1/(α−1).
• For any α the quantity r1−α falls linearly to zero at tf = r01−α / ((α−1) A).
• Forecast: least-squares line through 1/r over the chosen share of the record, zero crossing −a/b.

Checks run in the page

• RK4 integration of the Voight law against the closed form.
• The exact inverse rate for alpha = 2 forecasting the true day.
• The linear quantity for the current alpha.
• An ensemble of noisy records for the unbiased slope.

Honest simplifications

Real swarms are not smooth and many accelerations stop without an eruption. The scatter here is a simple log-normal factor, there is no change of regime and no false alarm, and the exponent is the same for the whole run. It illustrates the method and its sensitivity, not a hazard forecast.

• Press Noisy counts and watch the forecast wander before it settles.
• Press Slower rise, alpha 1.5 to see the straight 1/r forecast come out late.
• Shorten the window to see the fit react faster but jump more.