Industry: Mining and Natural Resources Product: ModelRisk Application: Mine safety analysis
A deep underground gold mine in the Sudbury basin operates at 2,250 m depth, where in-situ stress reaches 62 MPa and the average uniaxial compressive strength of the host quartzite is 195 MPa. Three minor rockbursts in the prior 18 months had been treated as incidents. The deterministic risk register scored the hazard "low frequency, high impact" without a number. The probabilistic seismic-risk model put numbers on it: the annual probability of a magnitude ≥ 2.5 event in the active mining stopes averages 8.4%, but the distribution is sharply right-skewed — the median stope-year sits at 3.9% while the worst-decile P90 reaches 23.7%. That dispersion, not the mean, is what the deterministic register missed.
The right tail is the whole story. A flat Poisson model with no over-dispersion and no weak-zone jump-diffusion would peg the annual probability near the 5.1% baseline marked on the chart — a clean point estimate that hides the geotechnical reality. The negative-binomial model with the weak-shear-zone jump fans the distribution out: the same mine has a long upper tail of high-probability stope-years that the Poisson view simply cannot represent.
The probability of a rockburst in a given stope-quarter depends on the burst-prone-ness index (BPI) — the ratio of induced stress to rock-mass strength — combined with the rate of stress redistribution from adjacent extraction. The model uses:
Monte Carlo over a 30-year mine life produces the distribution of annual event probabilities and a per-year aggregate-loss distribution. The mean annual P(M ≥ 2.5 event) is 8.4%, and the mean probability of at least one such event over the 30-year mine life is 60% — not a remote contingency, but a more-likely-than-not exposure over the life of the operation. The Poisson-only point estimate of 5.1% per year invites exactly the "low frequency" framing the register reached for; the dispersion is what reframes the hazard.
Conditional on a magnitude ≥ 2.5 event, the loss distribution is the sum of:
Conditional expected loss per event = $14.2M; conditional P90 = $20M and P95 = $23M. Unconditional annual expected loss = 0.084 × $14.2M ≈ $1.2M. The expected-loss-only number radically understates the tail: in any given year, with 8.4% probability, the mine is exposed to a $14M+ event the budget has not provided for.
The deterministic risk register said "rockburst: low frequency, high impact, mitigations in place." The probabilistic model said annual expected loss $1.2M, 30-year cumulative probability of at least one M ≥ 2.5 event 60%, and conditional P95 loss $23M. The first description supports the existing $0.5M annual safety-capital budget; the second supports an $8M one-off destressing-and-monitoring program with expected NPV +$4.9M against the avoided losses.
The tornado is built on the 30-year aggregate-loss P95 (baseline ≈ $118M), not the mean — because the captive-insurance reserve is sized off the tail, not the average. Stress-concentration k at stope abutments leads, swinging aggregate loss by roughly ±$9M. Weak-shear-zone hit probability (the jump component) is second. Event over-dispersion (negative-binomial vs. Poisson) is third. Ground-support remediation cost is fourth.
Rockbursts are not "low-probability, high-impact" in any meaningful sense — over a mine life they are more likely than not, and the only honest planning answer is a distribution of consequences.