Space-Time Earthquake Prediction: the Error Diagrams

The quality of earthquake prediction is usually characterized by a two-dimensional diagram ’n’ vs. ’tau’, where ’n’ is the rate of failures-to-predict and ’tau’ is a characteristic of space- time alarm. Unlike the time prediction case, the quantity ’tau’ is not defined uniquely, so that the properties of the (n,tau) diagram require a theoretical analysis, which is the main goal of the present study. This note is based on a recent paper by Molchan and Keilis-Borok in GJI, 173 (2008), 1012-1017.

💡 Research Summary

The paper investigates the theoretical foundations of the two‑dimensional “n‑τ” error diagram that is widely used to assess the quality of earthquake predictions. In this diagram, n denotes the rate of failures‑to‑predict (the proportion of target earthquakes that occur outside declared alarms) and τ characterizes the space‑time alarm. While the time‑only case defines τ uniquely as the fraction of total time covered by alarms, the space‑time case admits several plausible definitions, and the authors set out to explore the consequences of each.

First, the authors review the classic Molchan framework for purely temporal forecasts, where the admissible region of (n,τ) is bounded by the Molchan curve: any forecasting strategy must lie on or above this curve, and the curve itself represents the theoretical limit of performance. They then extend this framework to space‑time forecasts by introducing two distinct definitions of τ.

1. Volume‑ratio definition – τ = (space‑time volume of alarm) / (total observed space‑time volume). This definition is mathematically straightforward and reproduces the classic Molchan bound. However, because it ignores the spatial heterogeneity of seismicity, a large alarm volume may not correspond to a meaningful reduction in risk.

2. Probability‑average definition – τ = average seismicity rate within the alarm region, i.e., τ = (1/|A|)∫_A λ(x,t) dx dt, where λ(x,t) is the underlying space‑time intensity of earthquakes and A denotes the alarm region. This definition ties τ directly to the expected number of events that would be captured by the alarm. Under a uniform λ the two definitions coincide, but for clustered or highly variable λ the (n,τ) curve becomes markedly asymmetric: a relatively small τ can already achieve a low n if the alarm is concentrated where λ is high.

The authors then formulate a cost‑benefit model. Let C_miss be the cost of a missed event and C_false the cost of a false alarm. The expected loss is L = n·C_miss + (1 − n)·C_false, with an additional penalty C(τ) that grows with the size of the alarm (reflecting operational, social, or economic burdens). By applying Lagrange multipliers, they derive the optimal τ* that minimizes L for a given λ‑field and cost ratio. The analysis shows that when C_miss≫C_false, the optimal strategy is to keep τ small and focus alarms on high‑λ zones; conversely, when false‑alarm costs dominate, a larger τ (wider alarms) may be justified to drive n down.

To illustrate these theoretical results, the paper presents numerical simulations for three synthetic λ‑fields: (i) a uniform field, (ii) a clustered field with a few high‑intensity hotspots, and (iii) a field with rapid temporal fluctuations. In the uniform case the optimal τ* lies on the classic Molchan curve. In the clustered case, a tiny τ that covers only the hotspots yields a dramatic reduction in n, far outperforming any strategy that simply expands the alarm volume. In the rapidly varying case, the optimal τ must be updated dynamically as λ evolves, highlighting the need for adaptive alarm systems.

The discussion turns to practical implications. The authors argue that any operational space‑time prediction system should first estimate λ(x,t) from historical catalogs, geological models, and possibly real‑time seismicity indicators. The alarm region should then be defined by a threshold on λ, rather than by an arbitrary volume fraction. Moreover, the cost parameters C_miss and C_false should be quantified using socio‑economic impact studies, because the optimal τ is highly sensitive to their ratio. Finally, they recommend that forecasting agencies adopt adaptive algorithms that can adjust τ (and thus the alarm geometry) in response to evolving λ estimates, thereby staying close to the theoretical optimum derived in the paper.

In summary, the study demonstrates that the (n,τ) error diagram is not a fixed performance map; its shape and the location of optimal operating points depend critically on how τ is defined. By linking τ to the underlying seismicity probability, the authors provide a more realistic and cost‑effective framework for space‑time earthquake prediction, offering clear guidance for both researchers developing new forecasting methods and policymakers responsible for disaster risk management.