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Seismology formula solvers

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CEFGIMPRS

Cyclic Stress Ratio for Liquefaction

CSR=0.65 amaxg σvσv′ rd\mathrm{CSR} = 0.65\,\dfrac{a_{max}}{g}\,\dfrac{\sigma_v}{\sigma'_v}\,r_d

SeismologyThe seismic demand side of liquefaction assessment, from Seed and Idriss's 1971 simplified procedure: the average cyclic shear stress an earthquake puts into a soil element, normalised by the effective stress holding that element together.

Energy Ratio Between Two Magnitudes

E2E1=10 1.5 ΔM\dfrac{E_2}{E_1} = 10^{\,1.5\,\Delta M}

SeismologyWhat a difference in magnitude is actually worth in energy: about 32 times per whole unit, and exactly 1000 times per two units. The fact that turns "only one point higher" into the most misleading phrase in earthquake reporting.

Epicentral Distance from the S–P Interval

d=Δt1Vs−1Vpd = \dfrac{\Delta t}{\frac{1}{V_s} - \frac{1}{V_p}}

SeismologyHow far away the earthquake was, from one seismogram and no synchronised clock: the gap between the P and S arrivals divided by the difference of the two slownesses. The oldest trick in observational seismology and still the first one taught.

Fundamental Period of a Soil Site

T=4HVsT = \dfrac{4H}{V_s}

SeismologyThe period at which a soft soil layer over stiff rock resonates: four times its thickness divided by its shear wave velocity. The number that explains why identical buildings a block apart can fare completely differently.

Gutenberg–Richter Frequency–Magnitude Relation

log⁡10N=a−bM\log_{10} N = a - bM

SeismologyHow many earthquakes of magnitude M or larger a region produces: a straight line on a log-count axis. Fitted to a catalogue rather than derived from physics, and the a and b that come out belong to that catalogue and that region.

Impedance Contrast Amplification

A=ρ1Vs1ρ2Vs2A = \sqrt{\dfrac{\rho_1 V_{s1}}{\rho_2 V_{s2}}}

SeismologyHow much a shear wave grows when it passes from stiff rock into soft soil: the square root of the impedance ratio between the two. The conservation-of-energy reason that soft ground shakes harder than the rock beside it.

Moment Magnitude (Mw)

Mw=23(log⁡10M0−9.05)M_w = \dfrac{2}{3}\left(\log_{10} M_0 - 9.05\right)

SeismologyThe magnitude every agency now reports, defined directly from seismic moment by Hanks and Kanamori in 1979. Unlike the older scales it does not saturate, so it stays meaningful for the largest earthquakes there are.

P-Wave Velocity from Elastic Moduli

Vp=K+43GρV_p = \sqrt{\dfrac{K + \frac{4}{3}G}{\rho}}

SeismologyHow fast a compressional wave travels through an elastic solid: bulk modulus plus four-thirds of the shear modulus, over density, square-rooted. The first arrival on every seismogram, and the wave that gives the P its name.

Radiated Energy from Surface-Wave Magnitude

log⁡10E=4.8+1.5 Ms\log_{10} E = 4.8 + 1.5\,M_s

SeismologyThe classical estimate of the seismic energy an earthquake radiates as waves, from its surface-wave magnitude, with E in joules. A 1950s calibration that modern broadband measurements only loosely agree with — useful for scale, not for precision.

S-Wave Velocity from Shear Modulus

Vs=GρV_s = \sqrt{\dfrac{G}{\rho}}

SeismologyHow fast a shear wave travels: shear modulus over density, square-rooted. The slower of the two body waves, the one that does most of the shaking damage, and the single number site classification codes are written around.

Seismic Moment

M0=μADˉM_0 = \mu A \bar{D}

SeismologyThe physical size of an earthquake: the shear modulus of the rock, times the area of fault that broke, times the average slip across it. The only measure of earthquake size that carries real units, and the quantity every modern magnitude is computed from.