Earthquake Early Warning: Seconds Before Shaking

Timing. With epicentral distance D and source depth h, the straight path is R = √(D² + h²). City arrivals are tP = R/vP and tS = R/vS. Detection uses a separate source-to-triggering-station path. Alert receipt is station P arrival + processing + delivery. The signed warning margin is tS − tAlert; available advance warning is max(margin, 0). These times are conditional on an alert being issued. S onset is a timing proxy; it is not a forecast of peak or damaging shaking.

Blind zone and animation. The surface radius reached by S at alert receipt is √max[(vS tAlert)² − h², 0]. With a sufficiently deep source, a zero radius can mean there is no blind zone even above the hypocentre. P and S animation circles are intersections of spherical fronts with a flat surface, r(t) = √max[(vt)² − h², 0], not circles expanding at surface speed v from time zero. Their radius is zero until the front reaches the surface. The gray disk is fixed at the alert-receipt time. The circular window is a schematic map, with no geographic direction or basemap.

Omori inversion. The same straight-path approximation gives Δt = R(1/vS − 1/vP), hence R = vP vS Δt/(vP − vS). Epicentral distance is √(R² − h²); the measured interval must be at least h(1/vS − 1/vP). One interval gives a radius, not a unique epicentre.

Shaking. Si and Midorikawa (1999), attached source s2, printed p. 67 (file page 5), Table 6, constrained closest-fault-distance regression:
log10[PGV600/(cm/s)] = 0.58 Mw + 0.0038 h + d − 1.29 − log10[X + 0.0028 × 10^(0.5 Mw)] − 0.002 X,
where h and X are numerical distances in km and d is 0, −0.02 or +0.12 for crustal, interface or intraslab events. PGV600 refers to stiff ground with S-wave velocity about 600 m/s. Multiply by the selected site factor to estimate site PGV. The 1.41 factor for 400 m/s engineering ground is documented by GEM's implementation of the NIED method. Reference cases use GEM's published alternative-implementation fixtures.

Intensity proxy. I = 2.68 + 1.72 log10[site PGV/(cm/s)] is the Midorikawa–Fujimoto–Muramatsu (1999) correlation, also reproduced in Tajima and Hayashida (2018), Appendix Eq. 7. The continuous value is shown only as an interpretation within I = 4–7. It does not assign an official JMA intensity class or reproduce the filtered, duration-dependent instrumental-intensity calculation.

Source discrepancy. Tajima and Hayashida, Appendix Eq. 4, prints −0.0038D. The original Si–Midorikawa Table 6 has +0.0038; the original abstract describes stronger motion for deeper events, and GEM's independent implementation agrees. This calculator retains the positive coefficient. No published values were fitted.

• Educational hypothetical scenario, not a warning service. No live input, location lookup or connection to JMA, SASMEX or ShakeAlert.
• Homogeneous isotropic medium, straight rays, flat surface and a point source for travel time. P speed exceeds S speed. JMA's introductory 7/4 km/s values are defaults, not a regional velocity model.
• The triggering-station distance represents the station that completes detection. Processing includes confirmation, signal accumulation and any additional station/network delay. A single alert-receipt time is used across the map.
• Magnitude affects PGV only. No amplitude threshold, false-alarm probability, missed detection, rupture evolution or network coverage model decides whether an alert is sent.
• Ground-motion distance is the closest 3D distance to the rupture. If unknown, hypocentral distance is used explicitly as a point-source proxy; this can underestimate near-fault shaking for extended ruptures.
• Site factors are PGV ratios relative to 600 m/s reference ground. The 400 m/s preset uses ×1.41; a custom factor is a user assumption.
• Default delays of 2 s processing and 2 s delivery are illustrative. Animation time changes only the wavefront view.
• Physical fields and result cards use the viewer's metric/US conversion. Wave speeds are entered in m/s (or ft/s in US mode). PGV cards use m/s (or ft/s); the regression steps explicitly retain cm/s. The schematic's written blind-radius label remains in km.
• Alert delivery ends when the user/device receives the alert. Human reaction and protective-action time are excluded from alert receipt, warning margin and blind-zone calculations.

Use for teaching and comparing scenarios, never for real-time protective decisions, structural design or equipment shutdown control. Follow official warnings and local emergency guidance.

The Si–Midorikawa database comprises Japanese events with Mw about 5.8–8.3 and depths 6–120 km (s2, file page 1). Inputs permit timing studies at 0–150 km depth and 0–500 km epicentral distance; the tool flags empirical depth extrapolation and paths over 300 km. The latter is a caution threshold chosen for this simple straight-ray timing model, not its published calibration limit. Layering, refraction and Earth curvature can cause substantial travel-time errors. Different regions require regional ground-motion and travel-time models.

Ground-motion distance envelope. Si and Midorikawa (1999), attached s2, file page 3 / printed p. 65, selected records within 100 km for Mw ≤ 6.2, 150 km for Mw 6.3–6.5, 200 km for Mw 6.6 to below 7.0, and 300 km for Mw ≥ 7.0. The calculator applies the transitions at Mw 6.3, 6.6 and 7.0; intermediate entered magnitudes below a transition retain the lower distance limit. It compares the actual ground-motion distance X_fault with that envelope, using the hypocentral proxy when selected. An exceedance separately flags PGV and the intensity proxy as distance extrapolations; results remain available for educational exploration. This published data-selection envelope is distinct from the 300 km surface-distance caution used for straight-ray timing.

PGV is a median empirical estimate, not a guaranteed amplitude. Original Table 8 (s2, file page 5) gives log10 PGV scatter of about 0.23, roughly a factor of 1.7 for one standard deviation, before uncertainty in site factors and source geometry. This is not a complete confidence interval. The JMA intensity correlation adds its own scatter (about ±0.21 in the published relation); do not interpret displayed decimals as measurement accuracy. Outside I = 4–7, the interpretation card is withheld.

No basin resonance, surface-wave dispersion, nonlinear soil response, liquefaction, duration, frequency content, aftershocks, tsunami hazard or building response is calculated. P-wave motion can already be strong; peak motion can occur well after S onset. Real warning availability also depends on evolving event estimates, alert thresholds and communications reliability.