This is a real point-process (not scripted) seismicity plot. Every earthquake — mainshock or aftershock — can trigger its own children, and the decay rate of triggered events follows the empirical Omori-Utsu law:
n(t) = K / (t + c)^p
where t is time since the triggering shock, c is a short time offset (~0.05 d here) and p (0.8–1.5) controls how fast the sequence dies out — this simulator's Omori-p slider. Chaining Omori-Utsu recursively, so every aftershock can itself spawn aftershocks, is the ETAS (Epidemic-Type Aftershock Sequence) model used operationally by seismic agencies (USGS, JMA) for real-time forecasts.
Each event's expected number of "children" scales with its own magnitude m via a productivity law:
μ(m) = K · 10^(α(m − M_min))
the Productivity slider K. Child magnitudes are drawn from the Gutenberg-Richter distribution P(M≥m) ∝ 10^(−bm), so most aftershocks are small and large ones are exponentially rarer. The magnitude-vs-time scatter below is the standard way seismologists read a real sequence: dots climbing in count right after t=0, thinning out as the Omori curve (orange, right axis, log-scaled) decays. Bath's law is the empirical observation that the single largest aftershock in a sequence runs about 1.2 magnitude units below the mainshock (M₀ − 1.2), shown here as a live comparison against what the stochastic run actually produced.
- Magnitude M₀ — mainshock size; sets the initial productivity and the Bath's-law reference line.
- Omori p-value — how steeply the aftershock rate decays with time; higher p means a sequence that quiets down faster.
- Productivity K — how many aftershocks each event triggers on average; higher K means a longer, denser cascade.
- λ(t) readout — the model's own forecast rate for the next 24 hours, recomputed live from every active branch, exactly the number an operational aftershock advisory would publish.
- Direct children of M₀ — the number of first-generation aftershocks spawned straight from the mainshock; averaged over many runs it grows as 10^(0.8·ΔM) with mainshock magnitude, the productivity law's signature.
Real-world relevance: this is the same statistical machinery — Omori-Utsu decay plus ETAS branching, increasingly paired with machine-learning corrections trained on catalog data — behind the short-term aftershock probability forecasts issued after major earthquakes.