8.4 Magnitude & Frequency Relations
Earthquake Magnitude Scales
Earthquake magnitude provides a single number that characterizes the size of an event. Several magnitude scales have been developed historically, each measuring a different aspect of the seismic wavefield. All are logarithmic: each unit increase corresponds to a tenfold increase in measured ground-motion amplitude. The modern standard is moment magnitude (Mw), which is directly tied to the physics of the source.
ML — Local (Richter) Magnitude
Defined by Charles Richter (1935) for southern California earthquakes. Based on the maximum amplitude recorded on a Wood-Anderson torsion seismometer at a specified distance. The scale is calibrated so that an ML 3 earthquake produces a 1-mm trace amplitude at 100 km epicentral distance. ML saturates (underestimates true size) for events above about M 6.5 because the high-frequency waves it measures do not scale with fault area for large ruptures. Still widely used for local earthquake monitoring.
mb — Body-Wave Magnitude
Measured from the amplitude of short-period (~1 s) P-waves recorded at teleseismic distances (30°–90°). Defined by Gutenberg (1945). Useful for global seismicity catalogs and nuclear test detection. Saturates above mb ≈ 6.5 because the ~1-s period cannot capture the long-period energy of large ruptures.
MS — Surface-Wave Magnitude
Measured from the amplitude of Rayleigh waves at a period of ~20 s, recorded at teleseismic distances. Defined by Gutenberg (1945). Better suited for larger earthquakes than mb because the 20-s period samples more of the source. However, MS also saturates above about MS 8.2 and is unreliable for deep earthquakes (which excite weak surface waves). The mb:MS discriminant is used to distinguish earthquakes from nuclear explosions (explosions have anomalously high mb relative to MS).
Mw — Moment Magnitude
Defined by Hanks & Kanamori (1979). Derived from the scalar seismic moment M0, which is a direct physical measure of the source (product of rigidity, fault area, and average slip). Mw does not saturate at any magnitude because it is based on the total energy release, not a single frequency band. It is the only magnitude scale used for the largest earthquakes and is the preferred standard for modern seismology.
Moment Magnitude & Seismic Energy
Moment Magnitude Definition
\[ M_w = \frac{2}{3} \log_{10}(M_0) - 6.07 \]
where M0 is the scalar seismic moment in N·m. This definition was calibrated by Hanks & Kanamori (1979) so that Mw coincides with MS for moderate earthquakes (M 3–7). The constant −6.07 arises from the conversion between dyne·cm (the historical unit, where the constant is 10.7) and N·m. Each unit increase in Mw corresponds to a factor of 101.5 ≈ 31.6 increase in M0.
Seismic Moment
\[ M_0 = \mu \, A \, \bar{D} \]
where μ is the shear modulus (~30 GPa in the crust, ~70 GPa in the upper mantle), A is the fault rupture area, and D̄ is the average fault displacement. Units: N·m (= Pa · m² · m). The 2011 Tōhoku earthquake had M0 ≈ 5.3 × 1022 N·m (Mw 9.1), with A ≈ 500 × 200 km² and D̄ ≈ 10–20 m.
Energy–Magnitude Relation
\[ \log_{10}(E_S) = 1.5 \, M_w + 4.8 \]
where ES is the radiated seismic energy in joules (Gutenberg & Richter, 1956). Each unit increase in Mw represents a factor of 101.5 ≈ 31.6 times more energy. An Mw 8.0 earthquake releases ~1000 times more energy than an Mw 6.0. An Mw 4.0 releases ~6.3 × 1010 J, roughly equivalent to 15 tons of TNT.
Gutenberg–Richter Frequency–Magnitude Relation
One of the most fundamental empirical laws in seismology is the Gutenberg–Richter (GR) relation (1944), which describes the frequency of earthquakes as a function of magnitude in any sufficiently large region:
Gutenberg–Richter Law
\[ \log_{10}(N) = a - bM \]
where N is the cumulative number of earthquakes with magnitude ≥ M in a given time and region, a describes the overall seismicity rate (productivity), and b is the slope, describing the relative proportion of small to large events. Globally, b ≈ 1.0, meaning that for each unit decrease in magnitude, earthquakes become approximately 10 times more frequent.
The b-value is remarkably stable near 1.0 for tectonic seismicity worldwide, but systematic deviations occur:
- b > 1.0: Volcanic regions, geothermal areas, and swarm sequences (high proportion of small events)
- b < 1.0: Regions of high differential stress, subducting slabs, and some fault segments before large earthquakes
- b ≈ 1.5: Induced seismicity (hydraulic fracturing, wastewater injection)
- b ≈ 2/3: Theoretical prediction for a system controlled by seismic moment release (as opposed to fault area)
Maximum Likelihood b-Value (Aki, 1965)
\[ b = \frac{\log_{10}(e)}{\bar{M} - M_{\min}} \]
where M̄ is the mean magnitude of the sample, Mmin is the minimum magnitude of completeness (the smallest magnitude above which all events are detected), and log10(e) ≈ 0.4343. This estimator is preferred over least-squares regression because it properly accounts for the discrete, binned nature of magnitude data and provides unbiased estimates with known standard errors.
Aftershock Statistics
Aftershocks are the most predictable aspect of earthquake behavior. Their temporal decay and magnitude distribution follow remarkably simple empirical laws:
Omori's Law (1894)
The rate of aftershocks decays as a power law in time following the mainshock:
\( n(t) = \frac{K}{(t + c)^p} \)
where n(t) is the aftershock rate at time t after the mainshock, K is a productivity constant proportional to mainshock magnitude, c is a small time offset (typically 0.01–1 day, partly reflecting catalog incompleteness), and p ≈ 1.0–1.2 is the decay exponent. For p = 1, the cumulative number grows logarithmically with time.
Båth's Law (1965)
The magnitude of the largest aftershock is, on average, about 1.2 magnitude units below the mainshock magnitude:
\( M_{\text{largest aftershock}} \approx M_{\text{mainshock}} - 1.2 \)
This is a statistical average with significant scatter (± 0.5). The 2011 Tōhoku mainshock (Mw 9.1) had a largest aftershock of Mw 7.9, a difference of 1.2, perfectly matching Båth's law. The relation arises naturally from the GR distribution applied to the aftershock population.
Seismic Hazard & Recurrence
Probabilistic seismic hazard analysis (PSHA) combines earthquake recurrence statistics with ground-motion prediction equations to estimate the probability of exceeding a specified level of shaking at a site. The simplest recurrence model treats earthquakes as a Poisson process (memoryless, random in time):
Poisson Probability of Exceedance
\[ P = 1 - e^{-t/T} \]
where P is the probability of at least one event occurring in time window t, and T is the mean return period. For a fault with T = 200 years and a 50-year exposure period: P = 1 − e−50/200 ≈ 0.22 (22% probability). For PSHA, the standard design criterion is the ground motion with a 2% probability of exceedance in 50 years, corresponding to a return period of ~2475 years.
| Magnitude (Mw) | Global/yr | Approx. Return Period | Typical Energy (J) |
|---|---|---|---|
| ≥ 8.0 | ~1 | 1 yr | 6.3 × 1016 |
| ≥ 7.0 | ~15 | ~1 month | 2.0 × 1015 |
| ≥ 6.0 | ~120 | ~3 days | 6.3 × 1013 |
| ≥ 5.0 | ~1,300 | ~7 hours | 2.0 × 1012 |
| ≥ 4.0 | ~13,000 | ~40 min | 6.3 × 1010 |
| ≥ 3.0 | ~130,000 | ~4 min | 2.0 × 109 |
Frequency estimates based on global GR statistics (USGS/NEIC catalog); energy from Gutenberg & Richter (1956) relation.
Notable Earthquakes: Source Parameters
The table below lists selected major earthquakes with their seismic moment, moment magnitude, and rupture dimensions, illustrating the scaling relationships discussed above:
| Earthquake | Mw | M0 (N·m) | L × W (km) | D̄ (m) |
|---|---|---|---|---|
| 1960 Chile | 9.5 | 2.7 × 1023 | 1000 × 200 | ~24 |
| 1964 Alaska | 9.2 | 8.2 × 1022 | 800 × 250 | ~14 |
| 2004 Sumatra | 9.1 | 4.0 × 1022 | 1300 × 150 | ~11 |
| 2011 Tōhoku | 9.1 | 5.3 × 1022 | 500 × 200 | ~15 |
| 1906 San Francisco | 7.9 | 6.0 × 1019 | 470 × 15 | ~4.5 |
| 1994 Northridge | 6.7 | 1.2 × 1019 | 18 × 24 | ~1.3 |
| 2023 Turkey (Pazarcık) | 7.8 | 5.4 × 1020 | 300 × 25 | ~4.0 |
Sources: Global CMT catalog; USGS finite-fault models; individual published studies.
Characteristic vs. Gutenberg–Richter Recurrence
For individual faults, two competing models describe how earthquakes recur:
Characteristic Earthquake
A fault segment repeatedly ruptures in events of approximately the same size at quasi-regular intervals. The magnitude–frequency distribution shows an excess of events near the maximum magnitude relative to the GR extrapolation from small events. Supported by paleoseismic evidence on some well-studied faults (e.g., Wasatch, San Andreas at Wrightwood). The recurrence interval T = D̄/vslip provides a first-order estimate.
Gutenberg–Richter on Individual Faults
The GR power law applies even at the single-fault level, with no preferred event size. All magnitudes up to the maximum are possible, with frequency governed by log10(N) = a − bM. This view implies that the “characteristic earthquake” is a sampling artifact — we simply have not observed long enough to see the full range of event sizes. Supported by numerical fault simulations and some statistical analyses of seismicity catalogs.
In practice, modern seismic hazard models (e.g., the USGS National Seismic Hazard Model) use a hybrid approach: characteristic events on mapped faults for the largest magnitudes, with GR-distributed background seismicity for smaller events in areal source zones. This captures both the fault-specific recurrence behavior and the diffuse seismicity that occurs away from known faults.
Key Takeaways
- Moment magnitude Mw = (2/3)log10(M0) − 6.07 is the non-saturating, physics-based magnitude standard
- Each Mw unit corresponds to ~31.6× more energy and ~10× fewer events
- The Gutenberg–Richter relation log10(N) = a − bM, with b ≈ 1.0, is a universal scaling law
- Omori's law describes aftershock rate decay: n(t) = K/(t+c)p, with p ≈ 1.0–1.2
- Båth's law: the largest aftershock averages 1.2 magnitude units below the mainshock
- Seismic hazard uses Poisson statistics: P = 1 − exp(−t/T) for the probability of exceedance
- Both characteristic earthquake and GR models inform modern hazard assessments through hybrid approaches