A Review on Scaling of Earthquake Source Spectra

  • Jeen-Hwa WangEmail author


In this review paper, the theoretical and observational studies of scaling of earthquake source displacement spectra (abbreviated as source spectra) are compiled and discussed. The earlier studies, including the kinetic models proposed by several authors [including Haskell (Bull Seismol Soc Am 56:125–140, 1966) and Aki (J Geophys Res 72:1217–1231, 1967), provided the so-called ω−1, ω-square, and ω-cube models. Aki (1967) favored the ω-square model and also assumed constant stress drop, Δσ, and self-similarity of earthquakes. Some observations agree to one of the two models, but others do not. Hence, numerous alternative forms of the three scaling models to interpret the observations have been made by seismologists. For the corner frequencies, fc, the seismic moment Mo scales with fc in the form of Mo ~ f c −3 . For fcP of the P-waves and fcS of the S-waves, fcP > fcS is more reasonable than fcP < fcS. Mo scales as T D 3 , where TD is the duration time of source rupture and inversely related to fc, for both small and large events. The second corner frequency or cut-off frequency, fmax, at higher ω, that exists in the source spectra could be yielded by source, path, and site effects. The mechanisms to cause the patch corner frequency are still open. Analytical and numerical studies of the scaling laws based on the dislocation (e.g., Aki 1967), cracks (e.g., Walter and Brune in J Geophys Res 98(B3):4449–4459, 1993), spring-slider (e.g., Shaw in Geophys Res Lett 20:643–646, 1993), and statistical physics models (e.g., Hanks in J Geophys Res 84:2235–2242, 1997), including self-organized criticality (e.g., Bak et al. in Phys Rev Lett 59:381–384, 1987), show different scaling laws under various physical conditions.


Scaling Earthquake source displacement spectra (or source spectra) Dislocation model Crack model Statistical physics model Spring-slider model Corner frequency fmax Duration of rupture Seismic moment 

List of symbols


(x, y, z)

Rectangular coordinate system

(r, θ, φ)

Spherical coordinate system


Position vector from the nucleation point to a observational point (= (r, θ, φ))


Hypocentral distance (r = |ȓ|)


Position vector from the nucleation point to a point on a fault plane (= (x, y, 0))


Distance from the initial rupture point to a point on a fault plane (rs = |ȓs|)




Convolution between two functions


Fourier transform


Fourier transform of a function w(t)


Fourier amplitude spectrum


Fractal dimension


Root-mean-squared (RMS) value of w(x) calculated in a range of x

B, C, CP, CS, C, E, ε, λ, ξ, n

Constants or scaling exponents


The first-order Bessel function


The step function (= 0 when t < 0 and = 1 when t ≥ 0)


F(U, V)

General friction force being a function of slip X or velocity V

F(V) = Fo(1 − V/Vc)

Linearly velocity-weakening friction


Static friction force


Dynamic friction force




Characteristic velocity

f(u, v)

Normalized general friction force of F(X, V)


Normalized linearly velocity-weakening friction force of F(V) (= 1 − v/υ)


Normalized velocity of V


Normalized characteristic velocity of Vc

Geometry and Kinematic Parameters of Faults and Earthquake Sources


Fault length (km)


Effective fault length (km)


Length of an asperity (km)


The radius of a circular crack (km)


Fault depth or width (km)


Seismogenic-zone depth (km)


Fault area (= LW) (km2)


A parameter with a dimension of length (L, W, or, R)

D(ȓs, t)

Slip (or displacement) at a point ȓs = (x, y, 0) on a fault plane (m)


Average slip (or displacement) on a fault plane (m)


Dip angle of a fault (°)


Critical weakening velocity (m/s)


Rupture velocity along a fault plane (m/s)


x axis component of vR (m/s)


y axis component of vR (m/s)

vR i

Initial rupture velocity


Average rupture velocity


Rise time of source rupture (s)


Duration time of source rupture (s)


Difference between TD and the coherent (average) rupture time


RMS acceleration of strong ground motions


Width of the displacement P pulse


Constant relative particle velocity over the ruptured area


Relative particle velocity at the center of the elliptic crack


Earthquake magnitude in general


Body-wave magnitude


Surface-wave magnitude


Seismic moment (Nm)


Moment magnitude


Seismic radiation energy (J)


Scaled energy (= ER/Mo)


Distance from the nucleation point to a point on a fault plane (= (x, y, 0))


Distance from the nucleation point to a point on the periphery of the final fault


The frequency–magnitude relation

Earthquake Source Spectra


Angular frequency (Hz)


Frequency (= ω/2π) (Hz)


Period (= 1/f) (s)


Wavenumber (m−1)


Corner angular frequency (Hz)


Corner frequency (= ωc/2π) (Hz)


Corner frequency obtained from the P-waves (Hz)


Corner frequency obtained from the S-waves (Hz)

fc1, fc2, and fc3

The multiple corner frequencies used by some authors


The maximum frequency (Hz)


The effective bandwidth of a spectrum

U(ȓ, t)

Displacement function at ȓ


Fourier transform of u(ȓ, t), i.e., F[U(ȓ, t)]

P(ȓ, α, β, t)

Path effect of seismic waves


P-wave velocity


S-wave velocity


A notation for either α or β


A ratio of vR to either α or β (= vR/c and c = α or β)


Attenuation factor of waves


Average density of the materials along the raypath (kg/m3 or g/cm3)


Rigidity of the materials along the raypath (N/m2)


RMS average of the radiation pattern


Fourier transform of P(ȓ, α, β, t), i.e., F[P(ȓ, α, β, t)]

S(ȓs, t)

Source slip function

A(ω) or A(f)

Fourier transform of S(ȓs, t), i.e., F[S(ȓs, t)]

ϕ(x, t)

A function to represent either Dt(x, t) or Dtt(x, t)

AC(ζ, τ)

Autocorrelation of ϕ(x, t)


Amplitude of AC(ζ, τ) when ϕ(x, t) = Dt(x, t)


Amplitude of AC(ζ, τ) when ϕ(x, t) = Dtt(x, t)

Φ(κ, ω)

Fourier transform of AC(ζ, τ)


Correlation time in Aki’s model (Hz−1)


Correlation length in Aki’s model (m−1)


Spectral amplitude at ω = 0


Ao for P-waves


Ao for S-waves


A function used in Brune’s source model (é = a fraction of Δσ)


A symbol to denote 2.21β/R used in Brune’s source model


A symbol to denote (ωL/2)[1/vR − cos(θ)/c] (c = α or β)

Statistical Physics Model


Self-organized criticality

Spring-Slider Model


Number of sliders


Slip of the ith slider measured from its initial equilibrium position


Velocity of the ith slider (= ∂Ui/∂t)

Fi(Ui, Vi)

Slip- and/or velocity-weakening frictional force on the ith slider

Fs i

Static frictional force on the ith slider


Dynamic frictional force on the ith slider


Mass of a slider (kg)


Coil-spring strength between one slider and the other (N/m)


Coil-spring strength per unit area (= Kc/δyδz) (N/m3)


Leaf-spring strength between a slider and the moving plate (N/m)


Leaf-spring strength per unit area (= Kl/δyδz) (N/m3)


Stiffness ratio (= Kc/Kl = kc/kl)


Space between two sliders


A limit of “sa” when a approaches zero and being the normalized rupture velocity and equal to vR/Doωo


The moving plate speed (≈ 10−10 m/s)


The driving force on a slider due to the moving plate (Newton)


Areal density (= m/δyδz)

fi(ui, vi)

Friction force per unit area on the ith slider (= Fi/δyδz = fof(ui, vi))


Characteristic slip distance of a slider exerted by a force Fo (or fo) through a spring with strength of Kl (or kl) (= Fo/Kl = fo/kl)


Predominant angular frequency of a single spring-slider system (= (Kl/m)1/2 = (kl/ρA)1/2) (Hz)


Ratio of ωo to υ(= ωo/u)


A ratio of ωo to 2υ (= ωo/2υ)


Normalized time scale (= ωot)


Normalized x axis coordinate(= x/Df)


Normalized displacement (= U/Df)


Normalized velocity (= du/dt)


Complementary solution of u


Particular solution of u


A symbol to denote (1 − 1/4υ2)1/2


A symbol to denote (4υ2 − 1)1/2


Fourier transform of uc


Fourier transform of up


Modeled seismic moment



Initial stress (strength) (MPa)


Final stress (MPa)


Effect stress (MPa)


Average stress (= (σo + σ1)/2) (MPa)


Static stress drop (= σo − σ1) (MPa)


Dynamic stress drop (MPa)


Tectonic stress at depth y (MPa)

Δσp(x, y)

Extra stress at (x, y)


Fraction of Δσ


Gravity acceleration (= 9.8 m/s2)



This review paper is dedicated to passed-away Profs. Keiti Aki and Leon Knopoff who published the ω-square model of source spectra and the 1-D spring-slider model, respectively, in 1967 (more than 50 years ago). They both made great contributions to seismology and also gave me numerous significant, useful advices and suggestions on my research. I would like to express my deep thanks to an anonymous reviewer for valuable comments and suggestions to help me to substantially improve this article. This study was financially supported by Academia Sinica, the Ministry of Science and Technology (Grant No. MOST-106-2116-M-001-005), and the Central Weather Bureau (Grant No. MOTC-CWB-107-E-02), Taiwan.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Institute of Earth SciencesAcademia SinicaNangang, TaipeiTaiwan

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