Seismic source spectra, stress drop and radiated energy, derived from cohesive-zone
models of symmetrical and asymmetrical circular and elliptical ruptures Yoshihiro Kaneko, GNS Science (New Zealand)
Peter Shearer, UC San Diego
seismic moment
τp
a
circular fault
a
0.3 MPa
Allmann and Shearer (2009)
circular fault
τo
w
source radius
S-wave speed
Corner freq.
of spectra
4 MPa
This study
Madariaga (1976)
Δσd
τd
Analysis of seismic spectra of
~2000 global events
slope =
A
Stress drop of
a circular fault
(Eshelby, 1957)
self-similarity?
50 MPa
Dynamic model of circular fault: classical problem revisited
Shear stress
Allmann and Shearer (2009)
Standard method for estimating stress drop from source spectra
Stress drop of an earthquake estimated from source spectra
Vrt
Distance
Model of Madariaga (1976):
Two issues addressed in this talk:
•  Brune (1970): A simple, somewhat ad-hoc model of a
circular fault, with infinite rupture speed Vr
•  Sato & Hirasawa (1973): analytical solution for a singular
crack model with instantaneous healing of slip, Vr = 0.9β
•  Madariaga (1976): dynamic calculations for a singular crack
with spontaneous healing of slip using FDM, Vr = 0.9β"
(1) Absolute value of stress drop estimated from source spectra
(2) Variability of estimated stress drops and radiated energy
−1.0
−1.0 −0.5
0.0
x/a
Fourier transform
0.5
y/a
−0.5
−1.0
1.0
−1.0 −0.5
0.0
x/a
0.5
0
0
P corner
frequencies
R1
Amplitude
Corner frequencies, fall-off rates
Amplitude
grid search
0.5
1.0
far-field displacement
Seismic spectra
1
4
P-wave spectra
R1
0
10
−1
10
−2
10
−3
10
−2
10
−1
0
1
10
10
10
Frequency (fa/β)
S corner
frequencies
R1
fC
corner
frequency
0.50
• 
For a model with the smallest cohesive-zone,
there are 305 node points along the source radius.
• 
• 
Part 2: Factors contributing to variability of estimated stress
drops and radiated energy
0.40
0.30
0.25
A’w = 11
0.20
1.7 MPa
A’w = 21
A’w = 42
A’w = 168
A’w = 84
0.15
0.10
2.3 MPa
A’w = 5.3
0
0.38
0.05
0.10
0.15
Mean fracture energy (Gμ/Δσd2 a)
for P wave;
0.26
kP = 0.29
R2
S corner
frequencies
R1
Abercrombie (2015)
0.20
Allmann and Shearer (2009)
for S wave
Application of the Madariaga model
overestimates stress drops by a factor of 1.7.
Spherical average of fc is larger by about 20% than that of Madariaga (1976).
k is smaller than the model of Sato & Hirasawa (1973) with kp = 0.42 and ks = 0.29.
R1
EQs
0.35
As the weakeing rate (Aw) increases, the solution approaches a singular crack
model with spontaneous healing of slip and becomes independent of Aw.
P corner
frequencies
lower bound of absolute tectonic stresses?
30 MPa
Asymmetrical circular source with Vr = 0.9β
Baltay et al. (2011)
Scaled energy
Implication for absolute levels of stress drops
P wave
S wave
0.32
0.21
0.45
Kaneko and Shearer (GJI, 2014; JGR, 2015)
Frequency
Expanding rupture on a circular fault
Constant rupture speed
Prescribed dynamic strength drop
Abrupt change of stress at the rupture front (i.e., singular crack model)
Our model: cohesive zone that prevents a stress singularity
~5000 CPU hours for the
highest resolution case
fall-off rate
Time
Allmann and Shearer (2009)
1 2 3
Time (tβ/a)
Normalized amplitude
0.0
−1
−2
−3
Normalized amplitude
−0.5
0.0
P-wave displ.
R1
Spherical average of corner freq. (fkc a / β)
Far-field body-wave displacements
0.5
2
1
0
0.5
1.0
Mode II
3
Stress change (Δσ/Δσd)
representation theorem
(Aki and Richards, 2002)
0.5
0.0
1.0
1.5
Slip (2uμ/Δσda)
Slip-rate functions at source points
y/a
dynamic rupture simulations
A factor of 5.5 difference in Δσ
between Brune and Madariaga
Vr = 0.9β"
Mode III
1.0
Assumed k from
Madariaga (1976)
Variation of corner frequencies over the focal sphere
Recipe for computing corner frequencies
Source model
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• 
• 
Well-recorded repeating EQs near SAFOD:
Δσ = 25 – 65 MPa for Madariaga model
Δσ = 15 – 40 MPa for Kaneko and Shearer
(2014) model
Asymmetrical elliptical source with supershear rupture Vr = 1.6β
kS = 0.28
P corner
frequencies
0.8
R2
R1
0.6
kP = 0.42
R2
S corner
frequencies
R1
kS = 0.33
0.8
R2
0.6
f’C
f’C
0.4
0.4
rupture
propagation
direction
rupture
propagation
direction
0.2
Asymmetrical circular source with subshear rupture Vr = 0.9β
Symmetrical circular source with Vr = 0.9β
P corner
frequencies
kP = 0.38
R1
S corner
frequencies
0.2
P corner
frequencies
kS = 0.26
R1
0.8
R1
0.6
Asymmetrical circular rupture
Symmetrical elliptical rupture
Asymmetrical elliptical rupture
To what extent the variability in seismically estimated stress drops and scaled energy
comes from differences in source geometry, rupture directivity and rupture speeds
Estimation of scaled energy (Esr/Mo)
Conclusions
Non-dimensional scaled energy:
1.0
0.8
0.6 E ’
r
0.4 Mo’
rupture
propagation
direction
Symmetrical
0.2
rupture
propagation
direction
Asymmetrical
•  The asymmetrical model displays a strong azimuthal dependence of fc due to larger
directivity effect
•  The spherical average of the S-wave fc is comparable to that of the symmetrical model
0
Asymmetrical
supershear rupture
•  The asymmetrical circular and elliptical models show large variations in the
estimated scaled energy; e.g., for supershear case, Esr/Mo ranges from 0.04 to 4.0
•  A factor of 5 difference in the spherical average of Esr/Mo is obtained from a variety
of source scenarios (0.6β<Vr<1.6β)
•  We have re-visited the classical problem of a circular fault and derived
a new relation between a source dimension and the spherical average
of corner frequencies of far-field body wave spectra.
•  In observational studies that assumed Madariaga (1976), the mean
value of Δσ may have been overestimated by a factor of 1.7.
•  At least a factor of 2 difference in the spherical average of fc is
expected in observational studies simply from variability in source
geometry, rupture directivity, and rupture speeds, translating into a
factor of 8 difference in estimated Δσ.
•  At least a factor of 5 difference in scaled energy is expected from the
variability in the same source characteristics. These numbers increase
with an insufficient station coverage (not discussed in this talk).
•  Mach waves generated by supershear rupture lead to much higher fc
and scaled-energy estimates locally, suggesting that supershear
earthquakes can be identified from the analysis of fc and scaled energy.
Kaneko and Shearer (GJI, 2014; JGR, 2015)
R1
kS = 0.28
0.8
R2
0.6
0.4
0.4
Symmetrical circular rupture
R2
S corner
frequencies
f’C
f’C
0.2
kP = 0.29
rupture
propagation
direction
• 
• 
• 
• 
0.2
P-wave fc are larger than those of subshear rupture
The pattern of the S-wave fc is different from that of subshear rupture
This is caused by Mach waves; the S-wave fc is largest at the Mach angle
A factor of 2 difference is obtained from a variety of source scenarios (0.6β<Vr<1.6β)