余震確率計算機

Calculate the probability and expected number of aftershocks following a mainshock.

Analysis

余震予測の科学

余震とは本震に続いて発生する、同じ断層域のより小さな地震です。地殻が新たなひずみ配置に適応する過程で、破壊された断層沿いおよびその周辺で応力が再分配されることにより発生します。余震の連続は数週間、数ヶ月、あるいは数年続くこともあります。2011年のM9.1東北地方太平洋沖地震では、10年以上にわたって顕著な余震が続きました。

この計算機はReasenberg-Jones（1989）の統計モデルを使用しています。これは2つの基本的な地震学の法則を組み合わせたものです。改良大森の法則は、余震の発生率が時間経過とともにべき乗則に従って減衰する様子を記述します。余震は本震直後に最も頻繁に発生し、急速に減少します。バースの法則は、最大余震が通常本震よりも約1.2マグニチュード単位小さいという統計的推定を提供します。これらのモデルを組み合わせることで、地震学者は特定のマグニチュード閾値を超える余震の確率と予想数を予測できます。

余震科学の主要概念

改良大森の法則（1894年）：余震発生率 n(t) = K / (t + c)^p。ここでtは本震後の時間、K, c, pは経験定数。発生率はおおよそ1/tの割合で減少します。
バースの法則（1965年）：最大余震は統計的に本震より約1.2マグニチュード単位小さくなりますが、例外もあり、本震に近いマグニチュードの余震が発生する場合もあります。
グーテンベルグ・リヒターのb値（通常約1.0）は、小規模余震と大規模余震の比率を表します：M5の余震1回に対し、約10回のM4余震が予想されます。
余震域は一般的に本震の断層破壊域に対応します。M7.0の地震では30〜50 kmの範囲で余震が発生する可能性があります。

主な用途

大地震後、損傷した建物に再入場しても安全かどうかの評価。
継続する余震の危険性を考慮した捜索・救助活動の計画。
教育・研究目的での余震減衰パターンの理解。
余震リスクが高い本震後の期間における保険リスクの推定。

How to Use

1
Enter Mainshock Parameters
Input the mainshock magnitude (Mw), date, time, and location. The Omori-Utsu law—the foundation of aftershock forecasting—requires the mainshock time as the reference point (t = 0) for the decay calculation.
2
Set the Forecast Window
Select the forecast time window (1 day, 1 week, 30 days) and the minimum magnitude threshold for the aftershock probability estimate. USGS operational forecasts use M3.0+ as the standard reporting threshold.
3
Review the Aftershock Forecast
See the expected number of aftershocks above your chosen magnitude threshold and the probability of at least one aftershock exceeding specific magnitude levels. Note that the forecast uncertainty increases with time since the mainshock.

About

Aftershock sequences are not random noise following an earthquake—they are systematic, predictable in a statistical sense, and carry important information about fault properties and regional stress fields. The scientific basis for aftershock forecasting traces to the Omori-Utsu law, which holds remarkably across tectonic environments from Japan to California to New Zealand. The ETAS (Epidemic Type Aftershock Sequence) model, developed by Ogata (1988), extends Omori-Utsu to capture the full clustering structure of seismicity: each earthquake (aftershock or mainshock) independently generates its own offspring sequence, creating a branching process. ETAS successfully reproduces the broad statistical features of seismicity catalogs and forms the basis of modern operational forecasting.

The completeness of the post-mainshock earthquake catalog is a critical practical limitation. In the hours immediately following a large earthquake, numerous small-to-moderate aftershocks occur whose seismographic coda overlap in time, preventing individual identification—a problem called 'catalog incompleteness.' The magnitude of completeness Mc rises sharply after a mainshock and decays over days to weeks, depending on network density and analyst processing capacity. This incompleteness affects parameter estimation in Omori-ETAS models and means early forecasts carry larger uncertainty. Modern approaches use template matching—cross-correlating continuous waveform streams with known event templates—to detect small aftershocks hidden within the coda, dramatically lowering Mc in the critical early hours.

Social communication of aftershock probabilities presents persistent challenges. Research by social scientists (e.g., Becker et al., 2019, GeoJournal) shows that probabilistic formats ('35% chance of M5+') are frequently misinterpreted by the public as certainties or dismissals. Newer guidance from USGS and GNS Science emphasizes communicating aftershock information in terms of what people should do, not just probabilities: inspect your home before re-entering; assume any aftershock large enough to feel is large enough to collapse weakened structures; follow official guidance rather than individual seismicity monitoring apps whose algorithms and parameters may not align with operational systems.

FAQ

What is the Omori-Utsu law?

The Omori-Utsu law is an empirical relation describing the temporal decay of aftershock rates following a mainshock. Original formulation by Fusakichi Omori (1894) expressed aftershock rate as K/(t + c), where t is time after the mainshock and K and c are constants. Tokuji Utsu (1961) modified it to K/(t + c)^p, where the exponent p typically ranges from 0.9 to 1.5 across different tectonic environments. A p-value of 1.0 means that if 100 aftershocks occur on day 1, about 50 will occur on day 2, 33 on day 3, and so on. The law applies over remarkable time scales—years to decades—and is the foundation of the Epidemic Type Aftershock Sequence (ETAS) model used in operational aftershock forecasting by USGS, GNS Science, and other agencies.

How long do aftershocks typically last?

Aftershock sequences decay according to the Omori-Utsu law but have no strict termination point—they asymptotically approach background seismicity rates over time. As a practical rule, for a M7.0 mainshock, notable aftershocks (M3.0+) may continue at elevated rates for 6–12 months; for a M8.0, 2–5 years; for a M9.0, decades. The 2011 Tohoku earthquake's aftershock sequence remained elevated above background rates for at least 3 years afterward. The Kaikoura M7.8 earthquake (New Zealand, 2016) produced productive aftershock sequences on multiple fault strands lasting years. Aftershocks that themselves generate sub-sequences are properly called 'secondary aftershocks' and are handled by the stochastic branching structure of the ETAS model.

Can an aftershock be larger than the mainshock?

By definition, the mainshock is the largest earthquake in a sequence, so an aftershock cannot retrospectively exceed the mainshock—if a larger event occurs, it is reclassified as the new mainshock and all preceding earthquakes (including the former 'mainshock') become foreshocks. In approximately 5–10% of earthquake sequences, the initial large event is followed within days by a still-larger event. This is known as the 'foreshock problem' and creates practical challenges for official communication. The Omori-Utsu aftershock model cannot be unambiguously distinguished from a foreshock-mainshock sequence in real time. USGS operational forecasts acknowledge this explicitly: after a significant earthquake, there is always a non-trivial probability (typically 5–10% for M ≥ 5 mainshock, decreasing rapidly with time) that a larger event will follow.

What is the Bath's Law?

Båth's Law (Markus Båth, 1965) states that the largest aftershock in a sequence is typically about 1.2 magnitude units smaller than the mainshock, regardless of mainshock magnitude. This empirical observation means a M7.0 earthquake can be expected to produce a largest aftershock around M5.8; a M8.0 around M6.8. The rule has important practical implications: a M6.8 aftershock following a M8.0 mainshock can itself cause significant additional damage and is within the range that could kill people and damage weakened structures. Båth's Law is incorporated into USGS operational aftershock forecasts and into the ETAS model's magnitude-frequency parameters. The largest aftershock does not always occur on the first day—for the 2010 Darfield M7.1 earthquake, the largest aftershock (M6.3 Christchurch, February 2011) occurred 5 months later and caused 185 deaths.

How do operational aftershock forecasts work?

The USGS Operational Aftershock Forecast System (OAF), deployed following significant US earthquakes, uses the Epidemic Type Aftershock Sequence (ETAS) model and the Reasenberg-Jones model in parallel to produce probabilistic forecasts. The forecasts are updated continuously as new aftershocks occur and refine parameter estimates. A typical OAF statement specifies: the probability of one or more M5.0+ aftershocks in the next day (e.g., 35%), the expected number of M3.0+ aftershocks in the next week (e.g., 8–20), and the probability of an event exceeding the mainshock magnitude (e.g., 2%). The GNS Science aftershock forecasting system in New Zealand and the JMA aftershock probability system in Japan follow similar Omori-ETAS frameworks. All systems stress that aftershock forecasts are probabilistic, not deterministic, and uncertainty bounds are wide particularly in the hours immediately after a mainshock when the seismicity catalog is incomplete due to coda interference.