Ground Motion Prediction Equation Calculator
Estimate median peak ground acceleration and shaking intensity from earthquake magnitude and distance. Illustrative only.
📉 What is a Ground Motion Prediction Equation?
A ground motion prediction equation (GMPE), also called an attenuation relation, is a statistical formula that predicts how strongly the ground will shake at a site for an earthquake of a given size and distance. It is fitted by regression to thousands of strong-motion recordings, and it returns a median value of a shaking measure such as peak ground acceleration (PGA) together with a scatter that describes how far individual recordings stray from that median.
Seismologists and engineers use GMPEs in probabilistic seismic hazard analysis, in rapid shaking maps such as the USGS ShakeMap, in building-code development, in insurance and catastrophe modelling, and in teaching, where they show in one line how magnitude, distance and energy loss combine. Planners also use them for scenario studies, asking for example what a magnitude 7.0 on a nearby fault would do to a city 10 km away.
This calculator uses the classic Joyner and Boore (1981) relation for PGA, rewritten in the natural-log form ln(PGA) = c0 + c1 M - c2 ln(r) - c3 r. The model is valid for moment magnitudes of about 5.0 to 7.7 and distances up to about 370 km, based on western North American earthquakes. It is a deliberately simple, historic relation chosen for transparency. It is illustrative only and must not be used for engineering design, code compliance or safety decisions.
A common misconception is that PGA alone describes damage. It does not. Damage depends on the frequency content and duration of shaking, on soil conditions and on the structure. Modern hazard work uses the NGA-West2 family of models, which include site terms such as Vs30, fault mechanism, hanging-wall effects, basin depth and many spectral periods. Those models are far more capable than this simple form. The estimate here also feeds the Modified Mercalli Intensity class, using the same Wald et al. (1999) relation as the Earthquake Intensity (MMI) Estimator.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — Moderate Quake 20 km Away
M = 6.5 at d = 20 km (the default)
Example 2 — Large Quake Close to a City
M = 7.0 at d = 10 km
Example 3 — Smaller Quake 30 km Away
M = 5.5 at d = 30 km
Example 4 — Great Quake 100 km Away
M = 7.5 at d = 100 km
❓ Frequently Asked Questions
⚠️ Important Caution
This page is an educational illustration of how ground motion attenuates. Real seismic hazard analysis uses the full NGA-West2 ground motion models, which include site terms (Vs30 and basin depth), fault mechanism, hanging-wall and directivity effects, and spectral accelerations at many periods, combined with probabilistic hazard integration. Do not use this calculator for engineering design, building code compliance, insurance pricing or any safety decision.
🔗 Related Calculators
What is a ground motion prediction equation?
A ground motion prediction equation (GMPE), also called an attenuation relation, is a regression formula that predicts a ground-shaking measure such as peak ground acceleration from earthquake magnitude, distance and sometimes site conditions. It is fitted to recorded strong-motion data and returns a median value with a statistical scatter.
How do you estimate PGA from magnitude and distance?
This calculator uses Joyner and Boore (1981): log10(PGA in g) = -1.02 + 0.249 M - log10(r) - 0.00255 r, where r = sqrt(d squared + 7.3 squared) and d is distance in km. For M 6.5 at 20 km it gives about 0.164 g.
What are the c0, c1, c2 and c3 coefficients?
Converted to natural logs, the Joyner and Boore PGA relation has c0 = -2.349, c1 = 0.5733, c2 = 1 and c3 = 0.005872 per km, with h = 7.3 km. c0 sets the overall level, c1 scales with magnitude, c2 is geometric spreading and c3 is anelastic attenuation.
What is peak ground acceleration in g?
Peak ground acceleration is the largest acceleration recorded at a site during shaking, expressed as a fraction of standard gravity (1 g = 9.80665 m/s squared). A PGA of 0.1 g is about 0.98 m/s squared and is typically felt as strong shaking but usually causes only light damage to well-built structures.
How accurate is this PGA estimate?
Only roughly. The relation returns a median, and real recordings scatter around it by about 0.26 log10 units, a factor near 1.8 either way. Site soil, rupture directivity, fault type and depth are not included, so a single recording can differ from the prediction by a factor of two or three.
What magnitude and distance range is valid?
The Joyner and Boore (1981) relation was derived from western North American earthquakes with moment magnitude 5.0 to 7.7 and distances up to about 370 km. The calculator still runs outside that window but labels the result as an extrapolation, which should be treated with extra caution.
Can I use this for engineering design?
No. It is a simplified teaching tool. Real seismic hazard analysis uses modern NGA-West2 models (for example Boore et al. 2014 or Campbell and Bozorgnia 2014) with site terms such as Vs30, fault mechanism, hanging-wall effects and full probabilistic treatment, and building codes specify design values from those studies.
How does the MMI in the result get calculated?
PGA is converted to cm/s squared and passed through the Wald et al. (1999) relation MMI = 3.66 log10(PGA) - 1.66, the same one used by the Earthquake Intensity (MMI) Estimator. It is best constrained between roughly MMI V and IX, and values outside that band are flagged.
Why does PGA flatten at short distances?
Because the effective distance is sqrt(d squared + h squared) with h = 7.3 km, it never drops below 7.3 km. This stands in for the finite depth and size of the rupture, so a site 1 km from the surface trace gets similar shaking to one at 3 km.
How much does PGA change for each extra magnitude unit?
In this relation each extra unit of magnitude multiplies median PGA by exp(0.5733), about 1.77, at any fixed distance. So M 7.5 shakes about 1.8 times harder than M 6.5 at the same spot. Modern models show weaker scaling at large magnitudes near the fault (saturation), which this simple form does not capture.
What is the difference between PGA and MMI?
PGA is an instrumental measurement of ground acceleration at a point, while Modified Mercalli Intensity is a descriptive scale of felt shaking and damage from I to XII. They are linked by empirical conversion relations, but the same PGA can correspond to different intensities depending on frequency content, duration and building type.