Peak Ground Acceleration Calculator
Solve the simplified attenuation relation PGA = a · 10b·M / (R + c)d for any of its seven unknowns, with step-by-step math, a live earthquake cross-section schematic, an accelerogram preview, and companion intensity metrics including MMI, JMA intensity, and estimated PGV.
Magnitude enters exponentially
The 10b·M term means each unit of magnitude multiplies PGA by 10b. For b = 0.5, that's a factor of 3.16; for b = 0.6, a factor of 4.0. This exponential dependence is why the difference between an M 6 and an M 7 earthquake is so dramatic — the source amplitude at the site is 3–4 times larger.
Distance enters as a power law
The (R + c)d denominator controls geometric spreading and anelastic attenuation. For d = 1.5 and no offset, doubling the distance reduces PGA by a factor of 21.5 ≈ 2.83. The offset c keeps the denominator finite at small R, representing the finite size of the fault and preventing an unphysical infinite amplitude at the epicentre.
Empirical, not physical
This is an empirical attenuation relation, not a first-principles derivation. The coefficients a, b, c, d are fitted to observed PGA data from many earthquakes and stations. Different regions and tectonic settings (active crust, stable continent, subduction zone) have different coefficient values. The form here is one of the simplest families; real models add terms for site conditions, fault mechanism, and directivity.
PGA drives seismic design
Building codes use PGA and the corresponding elastic response spectrum to define the design earthquake. A site with 0.3 g PGA will typically require a much stronger structure than one with 0.1 g. The complete hazard assessment involves probabilistic analysis over all possible earthquakes, but the attenuation relation here is the core element of every such analysis.
How to Use This Calculator
The simplified attenuation relation links seven quantities: PGA, magnitude, distance, and four empirical coefficients. Pick which one you want the tool to solve, then fill in the others.
1. Choose what to solve for
Pick PGA, M, R, a, b, c, or d — the tool rearranges the relation automatically for whichever you choose. All seven are solved analytically.
2. Use realistic presets
PGA presets span 0.05 g (weak) to 1.0 g (extreme). Magnitude presets span 4.5 to 7.5. Distance presets span 5 km to 250 km. Coefficients use typical published values.
3. Read the result and steps
The result card shows the solved quantity with engineering notation. The step-by-step panel walks through the source term, the path term, and the division.
4. Check the earthquake cross-section
The diagram shows a fault cross-section with epicentre, hypocentre, ray path to the site, distance R, depth h, and a live accelerogram at the site with the current PGA. Below is the attenuation curve with the current position marked.
5. Switch to Advanced for depth
Advanced mode adds a chart of PGA vs. magnitude, a 1% sensitivity check, and companion metrics including PGA in m/s² and cm/s², estimated PGV, MMI, JMA intensity, and a classification of the shaking severity.
6. Cross-check with the reference table
The reference section lists typical PGA values for different magnitude–distance combinations and their corresponding MMI intensities.
What This Tool Offers
Built around the classical attenuation relation used in seismic hazard analysis and earthquake engineering.
Seven-way solving
Solve for PGA, magnitude, distance, or any of the four empirical coefficients — whichever the problem gives you.
Live earthquake cross-section
A fault cross-section with epicentre, hypocentre, ray path, site, and a live accelerogram showing the current PGA.
Dual comparison charts
PGA vs. distance at three magnitudes, plus PGA vs. magnitude at three distances.
Companion earthquake metrics
Advanced mode reports PGA in g, m/s², cm/s², estimated PGV, MMI, JMA intensity, and a classification of shaking severity.
Built-in sensitivity check
See how much PGA moves for a 1% change in each input. Magnitude has the strongest effect; distance has a milder one.
Reference tables
Typical PGA values for different M–R combinations and their corresponding MMI intensities, plus a lookup grid.
Reference Tables
Typical PGA values for different magnitude–distance combinations, plus a lookup grid.
PGA quick lookup (g) for typical attenuation coefficients
PGA = a · 10b·M / (R + c)d with a = 0.03 g, b = 0.5, c = 10 km, d = 1.5.
| M \\ R (km) | 5 | 10 | 30 | 50 | 100 | 200 |
|---|
PGA, MMI, and Shaking Severity
Approximate correspondence between PGA and Modified Mercalli Intensity (MMI). Values are representative and vary with site conditions.
| PGA (g) | PGA (m/s²) | MMI | Shaking severity | Typical damage |
|---|---|---|---|---|
| < 0.002 | < 0.02 | I | Not felt | None |
| 0.002 – 0.005 | 0.02 – 0.05 | II–III | Weak | None |
| 0.005 – 0.015 | 0.05 – 0.15 | IV | Light | Rattling of objects |
| 0.015 – 0.04 | 0.15 – 0.4 | V | Moderate | Minor damage |
| 0.04 – 0.10 | 0.4 – 1.0 | VI | Strong | Cracks in weak plaster |
| 0.10 – 0.25 | 1.0 – 2.5 | VII | Very strong | Damage to poorly built structures |
| 0.25 – 0.50 | 2.5 – 5.0 | VIII | Severe | Considerable damage in ordinary buildings |
| 0.50 – 1.00 | 5.0 – 10 | IX | Violent | Heavy damage, some collapse |
| > 1.00 | > 10 | X+ | Extreme | Widespread destruction |
Variable Reference
| Symbol | Meaning | Typical range | Units |
|---|---|---|---|
| PGA | Peak ground acceleration | 0.001 – 2.0 | g |
| M | Earthquake magnitude | 4.0 – 8.5 | Mw |
| R | Epicentral distance | 1 – 500 | km |
| a | Attenuation coefficient | 0.01 – 0.10 | g |
| b | Magnitude coefficient | 0.3 – 0.8 | — |
| c | Distance offset | 5 – 30 | km |
| d | Distance attenuation exponent | 1.0 – 2.5 | — |
What Does Peak Ground Acceleration Measure?
Peak ground acceleration is the maximum value of horizontal ground acceleration recorded at a site during an earthquake. It is expressed as a fraction of g (9.80665 m/s²), so an acceleration of 0.3 g means the ground moved with a peak horizontal acceleration of about 2.94 m/s². PGA is the single most-used measure of earthquake shaking intensity for engineering purposes: it is the input to seismic design codes, it is closely correlated with observed damage, and it can be estimated from magnitude and distance through an attenuation relation without requiring a full earthquake source model.
Attenuation Relation Formula
The simplified attenuation form used in this calculator is:
Where M is earthquake magnitude and R is epicentral distance. Rearranged to solve for magnitude or distance:
R = ( a · 10b·M / PGA )1/d − c
And for the four empirical coefficients:
b = log10[ PGA · (R + c)d / a ] / M
c = ( a · 10b·M / PGA )1/d − R
d = ln( a · 10b·M / PGA ) / ln( R + c )
Attenuation Variables
| Symbol | Meaning | Notes |
|---|---|---|
| PGA | Peak ground acceleration | Maximum horizontal ground acceleration |
| M | Earthquake magnitude | Commonly moment magnitude Mw |
| R | Epicentral distance | Horizontal distance from epicentre to site |
| a | Attenuation coefficient | Scale factor in units of g |
| b | Magnitude coefficient | Exponent on M in the source term |
| c | Distance offset | Keeps the denominator finite and models source size |
| d | Distance attenuation exponent | Controls how fast PGA decays with distance |
How Each Parameter Moves the PGA
PGA is extremely sensitive to magnitude and mildly sensitive to distance. A 1% change in M changes PGA by roughly b·ln(10) ≈ 1.15% (for b = 0.5) — but remember that a 1% change in magnitude is small compared to the range of magnitudes: going from M 6 to M 7 (a 16% increase) raises PGA by a factor of ~3.16. Distance has a stronger absolute effect at short range and a weaker effect at long range, since the attenuation is a power law. The coefficients a, b, c, d control the shape of the entire attenuation curve and are the elements fitted from observational data.
PGA Calculation Examples
Source term: a · 10b·M = 0.03 × 100.5·6.5 = 0.03 × 103.25 = 0.03 × 1778.28 = 53.35 g
Path term: (R + c)d = (30 + 10)1.5 = 401.5 = 252.98
PGA = 53.35 / 252.98 = 0.211 g
In m/s²: 0.211 × 9.80665 = 2.07 m/s²
MMI estimate: 3.66·log10(207) − 1.66 ≈ 6.75
10b·M = PGA · (R + c)d / a = 0.5 × 251.5 / 0.03 = 0.5 × 125 / 0.03 = 2083.33
b·M = log10(2083.33) = 3.3187
M = 3.3187 / 0.5 = 6.64
Frequently Asked Questions
Because earthquake magnitude itself is a logarithmic measure of source energy, and PGA is proportional to the source amplitude. The empirical scaling from source energy to PGA at a given distance is approximately a power law, which becomes exponential when written in terms of magnitude. The coefficient b captures the slope of that relationship; values around 0.5 mean that each unit of magnitude multiplies PGA by roughly √10 ≈ 3.16. This exponential dependence is why a magnitude 7 earthquake produces vastly stronger shaking than a magnitude 6, even at the same distance.
Epicentral distance R is the horizontal distance from the surface projection of the source to the site. Hypocentral distance Rhypo is the slant distance from the focus (hypocentre) to the site: Rhypo = √(R² + h²), where h is the focal depth. At long distances (R ≫ h) they are nearly identical. At near-field distances (R comparable to or smaller than h) they differ significantly, and the hypocentral distance is the physically correct measure of path length. Many attenuation models use Rhypo; the simplified form here uses epicentral R, which is adequate for R greater than about 20 km.
Because pure geometric spreading would give infinite amplitude at zero distance, which is physically impossible. The offset c keeps the denominator finite and represents three effects: (1) the finite rupture dimension of the fault, which means a large earthquake has its energy released over an extended source rather than a point; (2) the finite depth of the hypocentre, which prevents the site from being infinitely close to the source; and (3) the empirical observation that near-source ground motion saturates rather than diverging. Values of c typically range from 5 to 30 km in published attenuation models.
Damage thresholds in modern construction begin around 0.10–0.20 g. Well-built buildings in developed countries are designed for 0.2–0.5 g (or more in high-seismic regions like Japan, California, and Chile), while unreinforced masonry can be damaged at 0.05–0.15 g. Very strong shaking of 0.5–1.0 g occurs in the near field of large earthquakes and can cause widespread damage even to engineered structures. Ground motions above 1.0 g are recorded only in rare near-fault conditions and are usually accompanied by surface rupture and permanent ground deformation.
In practice, you choose a published attenuation model that has been fitted to data from a region similar to your study area. Examples include the Boore–Atkinson–Boore (BAB) models for western North America, the Akkar–Bommer model for Europe and the Middle East, the Zhao model for Japan, and the Campbell–Bozorgnia model for global active crustal regions. Each of these publishes its own coefficients. The simplified form here captures the essential physics but is not a substitute for a properly calibrated regional model; use it for educational and preliminary purposes, not for regulatory seismic hazard assessment.
PGA is the maximum ground acceleration at any frequency — the peak of the accelerogram. Spectral acceleration Sa(T) is the maximum acceleration experienced by a single-degree-of-freedom oscillator with natural period T. Structural design codes use a response spectrum that gives Sa as a function of period, and the plateau of that spectrum is typically about 2.5 times PGA for periods around 0.2–0.5 s. PGA is the zero-period anchor of the response spectrum and remains the single most useful scalar measure of shaking intensity for rapid hazard assessment and post-earthquake reconnaissance.
Variables
| Symbol | Quantity | Unit |
|---|---|---|
| PGA | Peak ground acceleration | g |
| M | Earthquake magnitude | |
| R | Distance from epicenter | km |
| a, b, c, d | Empirical regional attenuation constants |
What it means
Peak Ground Acceleration (PGA) attenuation relationships are empirical equations that predict the maximum ground shaking at a given distance from an earthquake. The simplified form uses constants a, b, c, d fitted to data. PGA decreases with distance and increases with magnitude. This is used in engineering seismology to design buildings, bridges, and infrastructure to withstand earthquakes. It is also used in hazard mapping and risk assessment. Understanding these relationships is essential for earthquake engineering and for ensuring public safety in seismic regions.
Worked example
Peak Ground Acceleration Attenuation – Two Detailed Examples
Real‑World| Parameter | Value |
|---|---|
| Magnitude (M) | 6 |
| Distance (km) | 10 |
| Parameter | Value |
|---|---|
| M | 7.5 |
| Distance | 50 |
Common mistakes
- PGA attenuation: PGA = a × 10^(b·M) / (R+c)^d – empirical ground motion prediction.
- Coefficients a,b,c,d: Region‑specific – use appropriate values from local ground‑motion models.
- Distance R: Usually hypocentral or source‑to‑site distance – in km.
- Magnitude M: Moment magnitude – not all scales are interchangeable.
- Units: PGA in g (acceleration of gravity) or cm/s² – check the model.
Applications
The simplified peak ground acceleration (PGA) attenuation relation, PGA = a · 10^(b·M) / (R+c)^d, estimates the expected ground shaking at a site from an earthquake of magnitude M and distance R. This is essential for seismic hazard analysis and for designing earthquake‑resistant structures. Engineers use attenuation relations to define the design ground motions for buildings, bridges, and infrastructure. The coefficients (a, b, c, d) are derived from empirical data and depend on the region. By predicting PGA, professionals can assess the vulnerability of structures, to plan emergency responses, and to develop building codes. Understanding PGA attenuation is critical for mitigating earthquake risk.
- Seismic hazard analysis and ground motion prediction
- Earthquake‑resistant design of buildings and lifelines
- Land use planning and risk management
- Development of building codes and design spectra
- Retrofit prioritisation of existing structures
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