Earthquake Magnitude Calculator

Introduction to local magnitude and the Richter scale

Every earthquake radiates elastic waves that a seismometer converts into a trace on a seismogram. Local magnitude, written M_L and still widely called the Richter scale, is the oldest surviving way of turning that trace into a single number. Charles F. Richter defined it in 1935 for southern California, and the definition was deliberately practical: measure the largest excursion on a Wood-Anderson torsion seismograph, correct it for how far the station sits from the source, and read off a logarithmic magnitude.

The scale is anchored by one arbitrary but famous choice. An earthquake that writes a peak trace amplitude of 1 mm on a Wood-Anderson instrument at a hypocentral distance of 100 km is defined to be magnitude 3.0. Everything else follows from an empirical attenuation curve that describes how that 1 mm reading would shrink or grow at other distances. Because the curve is logarithmic in amplitude, one whole magnitude unit corresponds to a tenfold change in ground motion at the station and to roughly a 32-fold change in radiated seismic energy.

This calculator does not use the loose classroom shortcut that many pages copy from each other. It implements the IASPEI standard local magnitude relation, the version recommended by the International Association of Seismology and Physics of the Earth's Interior for reporting M_L from digital data. That relation carries the modern southern California distance correction of Hutton and Boore (1987) and the measured Wood-Anderson static magnification of 2080, so the numbers it produces are directly comparable with the values a seismic network would compute for the same reading.

How to use this estimator with a real seismogram reading

The calculator needs one amplitude and one distance. Work through the inputs in order:

1. Measure the amplitude. On a horizontal component that has been deconvolved to ground displacement and reconvolved with the Wood-Anderson response at unit gain, take half of the maximum peak-to-peak excursion of the S-wave group. Enter that number and select its unit. Choose nanometres, micrometres or millimetres if you already have a ground-displacement figure; choose mm on a Wood-Anderson trace if you are reading a paper or simulated Wood-Anderson record directly, and the calculator will divide by the 2080 static magnification for you.
2. Choose how you know the distance. The equation wants hypocentral distance, the straight-line distance from the station to the focus. If your catalogue already gives it, select that option. If you have an epicentral distance and a focal depth, select the second option and the calculator combines them with Pythagoras. If all you have is a seismogram, select the third option and enter the S-minus-P arrival-time difference in seconds; the tool converts that into epicentral distance and then adds the depth.
3. Estimate the magnitude. The results panel reports M_L to two decimals, the descriptive band it falls in, the hypocentral distance actually used, a term-by-term breakdown of the equation, and a sensitivity table showing how the answer would move if your amplitude or distance were wrong.
4. Read the chart. The curve below the results shows M_L as a function of assumed hypocentral distance for the amplitude you entered, with your own case marked. It is the fastest way to see whether a mislocated epicentre would matter.
5. Keep or share the run. The address bar updates with every calculation, so the link reproduces the exact scenario. The buttons under the result copy a plain-text summary to the clipboard or download it as a .txt file for a lab notebook.

Inputs are validated as you calculate. Non-numeric or empty fields, zero or negative amplitudes, zero or negative distances, and hypocentral distances beyond the 1000 km validity limit of the relation all produce an explicit message instead of a number, so the panel never shows a stale or meaningless result.

The IASPEI local magnitude formula, term by term

Richter's original definition is a difference of logarithms. The magnitude is the logarithm of the measured amplitude minus the logarithm of the amplitude that a reference earthquake of magnitude zero would have produced at the same distance:

The whole difficulty of the scale lives in that second term, the distance correction or attenuation function. Richter published it as a table. Hutton and Boore re-derived it in 1987 from 7355 Wood-Anderson amplitudes recorded by the Southern California Seismic Network, and their curve is the one in use today:

Substituting that curve, converting the trace amplitude in millimetres into ground displacement in nanometres through the measured Wood-Anderson static magnification of 2080, and collecting the constants gives the compact form that IASPEI recommends and that this calculator evaluates:

with A the maximum horizontal ground-displacement amplitude in nanometres on a Wood-Anderson instrument simulated at unit gain, and R the hypocentral distance in kilometres. Each piece has a job:

• log₁₀(A) is the measurement. Ten times the ground motion adds exactly 1.00 to the magnitude, which is why the scale can span microearthquakes and great earthquakes with the same two-digit number.
• 1.11 log₁₀(R) is geometrical spreading. Surface-wave-like and direct S-wave energy spreads out as it travels, so a distant station under-reads and needs a positive correction that grows with the logarithm of range.
• 0.00189 R is anelastic attenuation. Real rock absorbs energy in proportion to path length, so a small term linear in distance is added on top of the geometric spreading.
• −2.09 is the calibration constant. It folds in Richter's anchor point and the conversion from a 2080-times-magnified trace in millimetres to ground displacement in nanometres.

The unit conversion built into the constant is worth stating explicitly, because it is where most published shortcuts go wrong:

When a catalogue supplies an epicentral distance and a focal depth rather than a hypocentral distance, the calculator combines them in the obvious way. Ignoring depth is the single most common source of error for shallow local networks, where a 10 km focus and a 12 km epicentral distance differ from the straight-line range by nearly 30 per cent:

If you only have a seismogram, the interval between the S and P arrivals fixes the epicentral distance. For a crust in which the P velocity is about 6.0 km/s and the ratio of P to S velocity is the Poisson value of the square root of three, the conversion factor works out near 8 km per second of S-minus-P time, the classic rule of thumb behind the Richter nomograph:

Finally, two ratios follow directly from the logarithmic definition and are reported alongside every result. The ground-motion ratio between two magnitudes is a power of ten, and the radiated-energy ratio is a power of ten with an exponent of one and a half, which is where the familiar factor of about 32 per magnitude unit comes from:

Worked example: a light event recorded 82 km away

A regional broadband station records an event. After the horizontal channel has been converted to ground displacement and filtered through the Wood-Anderson response, half of the maximum peak-to-peak S-wave amplitude measures 5400 nm, which is 5.4 micrometres of ground displacement and would be about 11.2 mm of pen excursion on a genuine Wood-Anderson drum. The network locates the event 82 km away in map view at a focal depth of 12 km.

1. Convert the epicentral distance and depth into a hypocentral distance:

   R = sqrt(82² + 12²) = sqrt(6868) = 82.87 km

2. Take the logarithm of the amplitude:

   log10(5400) = 3.7324

3. Apply the geometrical spreading term:

   1.11 × log10(82.87) = 1.11 × 1.9184 = 2.1294

4. Apply the anelastic attenuation term:

   0.00189 × 82.87 = 0.1566

5. Add the terms and subtract the calibration constant:

   ML = 3.7324 + 2.1294 + 0.1566 − 2.09 = 3.93

The station therefore reports M_L 3.93, a light earthquake that people close to the epicentre would clearly feel indoors but that would not be expected to damage well-built structures. Compared with an M_L 5.0 event, the ground motion at an equivalent station would be about 12 times smaller and the radiated energy about 40 times smaller.

It is worth checking the equation against its own anchor point, because a formula that fails its calibration is a formula that has been copied incorrectly. Richter defined magnitude 3.0 as a 1 mm trace amplitude at 100 km. One millimetre of trace at a gain of 2080 is 480.8 nm of ground displacement, so the equation gives log10(480.8) + 1.11 × log10(100) + 0.00189 × 100 − 2.09 = 2.6819 + 2.2200 + 0.1890 − 2.09 = 3.00. The relation reproduces its own definition to two decimal places, and you can confirm this in the calculator by selecting the Wood-Anderson trace unit, entering 1 mm, and setting a hypocentral distance of 100 km.

Interpreting the estimated magnitude

Magnitude is a dimensionless property of the source. It is not a prediction of shaking at any particular address, and it is not an intensity. Two events of identical magnitude can produce completely different experiences depending on depth, distance to population, sediment thickness and building stock. The Modified Mercalli Intensity scale, not magnitude, describes what people felt and what broke.

The descriptive bands below follow the classes the USGS uses in its public material. They are conventions for talking about earthquakes, not thresholds with physical meaning, and the boundaries are soft.

Because different magnitude scales measure different parts of the wavefield, a published number is only meaningful together with its type. The second table sets local magnitude beside the scales that agencies use for larger or more distant events.

Limitations, assumptions and honest error bars

The equation is empirical, regional and single-station. Every one of those words is a limitation:

• The attenuation curve is southern Californian. The coefficients 1.11 and 0.00189 were fitted to crustal paths in southern California. Stable continental interiors attenuate far less and subduction fore-arcs far more, so agencies elsewhere refit the linear term for their own region. Expect a regional bias of one or two tenths of a magnitude unit if you apply the standard coefficients far from their calibration area.
• It assumes a Wood-Anderson response. The relation is defined on a simulated Wood-Anderson instrument, which is a 0.8 s pendulum with damping near 0.7. Feeding it a raw broadband displacement peak, a velocity peak, or an unfiltered accelerogram will not give a comparable number.
• It saturates. A magnitude 7.5 rupture radiates most of its energy at periods far longer than the roughly 1 s that a Wood-Anderson instrument responds to, so the trace amplitude stops growing in proportion to the source. Above about M 6.5 local magnitude systematically under-reports size and moment magnitude should be used instead.
• One station is not a magnitude. Radiation-pattern lobes, site response and azimuthal path differences make single-station values scatter by a few tenths of a unit. Networks average many stations and both horizontal components and apply per-station corrections before publishing.
• Depth matters at short range. At an epicentral distance of 10 km, a 5 km focus and a 15 km focus differ in hypocentral distance by a factor of about 1.6, which is roughly 0.25 magnitude units. Omitting depth is a real error, not a rounding detail.
• Amplitude measurement is a judgement. The convention is half the maximum peak-to-peak S-wave excursion, but noisy records, clipped channels and overlapping events all make that reading ambiguous, and a factor-of-two disagreement in the reading is a 0.30 magnitude unit disagreement in the answer.
• Nothing here forecasts damage. Local site amplification, liquefaction, landslides, tsunami potential and structural vulnerability are outside the scope of any magnitude equation.

Treat the output as a defensible single-station estimate with an uncertainty of roughly plus or minus 0.2 to 0.3 magnitude units, and treat any published catalogue value from a national agency as authoritative in preference to it. If you are responding to a real earthquake, follow official guidance rather than a web calculator.

Frequently asked questions about earthquake magnitude