Energy scaling, saturation, and the moment magnitude unification — graduate level
01 — Magnitude Scales
ML, mb, Ms and Mw — Why Four Scales?
Each classical scale measures seismic wave amplitude at a specific period. Because source spectra have a corner frequency fc that decreases with earthquake size, large earthquakes shift energy below the measurement frequency — causing saturation.
M = log(A/T) + f(D, h) + C [generic form, amplitude A at period T, distance correction f]
ML
Local magnitude T ~ 0.1–1 s Wood-Anderson D < 600 km
Saturates ~M 7
mb
Body-wave mag. T ~ 1 s P-wave amplitude D = 20–100 deg
Saturates ~M 6.5
Ms
Surface-wave mag. T ~ 20 s Rayleigh wave D = 20–160 deg
Saturates ~M 8
Mw
Moment magnitude All periods From M⊂0; = μAD No saturation
No saturation
Each horizontal band marks the measurement period of a scale. As earthquake size grows, the corner frequency fc drops. Once fc < fmeasurement the scale saturates: the spectral plateau is no longer sampled.
02 — Moment & Energy
Seismic Moment, Radiated Energy & the 101.5 Rule
The Hanks & Kanamori (1979) relation ties seismic moment M0 to moment magnitude Mw. Each unit of Mw corresponds to a factor of 101.5 ≈ 31.6 in M0, and ≈ 31.6 in radiated energy Es. Drag the slider to feel the scale.
A Mw 9 earthquake releases ~1000x more energy than Mw 7, and ~106 times more than Mw 5. The entire annual global seismicity equals roughly one Mw 8 event.
M₀ & E_s vs Mw
Brune spectrum & source radius
03 — Saturation
The Saturation Mechanism — Corner Frequency & Spectral Sampling
For a Brune (omega-squared) source model, the corner frequency fc scales as M0-1/3. As Mw grows, fc drops below the measurement frequency of classical scales — the amplitude at that frequency stops growing even though M0 keeps increasing.
When fc drops below the measurement band (vertical dashed lines), the amplitude at that frequency saturates. Mw uses the low-frequency plateau (M0) and never saturates.
Source spectrum |U(f)|
f_c (corner freq)
ML band
mb band
Ms band
04 — Saturation Curves
Measured vs True Magnitude — Saturation Curves
Plotting classical magnitude against Mw reveals the saturation plateau. mb saturates around Mw 6.5, Ms around Mw 8. The dashed 1:1 line shows where a perfect unsaturated scale would lie. Drag the stress drop to see how it shifts the saturation onset.
Parameters
Stress drop Δσ 3.0 MPa
Higher stress drop shifts fc upward, delaying saturation to higher Mw. This is why intraplate earthquakes (higher Δσ) tend to saturate at slightly higher classical magnitudes than subduction-zone events.
ML
mb
Ms
1:1 (no saturation)
05 — Gutenberg-Richter
Gutenberg-Richter Relation & the b-value
The frequency-magnitude distribution follows log N = a − bM, with b ≈ 1 globally. Each unit increase in magnitude corresponds to ~10x fewer earthquakes. The b-value encodes the ratio of small to large events and is related to the heterogeneity of stress in the crust.
log N(M) = a − b·M | b ≈ 1.0 globally | b = (3/2) / [log(M0_max/M0_min)]
Parameters
b-value 1.0
a-value (productivity) 8.0
Completeness magnitude Mc 2.0
b < 1 indicates relatively more large events (high-stress environments). b > 1 indicates more small events (volcanic, induced seismicity). The maximum credible magnitude is bounded by tectonics, not by the G-R relation alone.