Mechanics of Earthquakes and Faulting

1
Mechanics of Earthquakes and Faulting
20 Apr. 2007
Earthquake Source Properties and Scaling
Hanks Paper (At least abstract, eqns 1-4,
Figures 1-3) Scaling laws
2
Earthquake Source Properties, Spectra, Scaling,
Self-similarity
Displacement and acceleration source spectra.
Spectra zero-frequency intercept (Mo), corner
frequency (?o or fc), high frequency decay (???),
maximum (observed, emitted) frequency fmax
?-square model, ?-2 ?-cube model, ?-3
?-n
Far-field body-wave spectra and relation to
source slip function
??
log u at R
Displacement waveform for P S waves
log freq. (?)
In general, very complex. ?(x, t) and ???? depend
on slip function, azimuth to observer and
relative importance of nucleation and stopping
phases
Aki, Scaling law of seismic spectrum, JGR, 72,
1217-1231, 1967. Hanks, b Values and ?-? seismic
source models implications for tectonic stress
variations along active crustal fault zones and
the estimation of high-frequency strong ground
motion, JGR, 84, 2235-2242, 1979.
3
Earthquake Source Properties, Spectra, Scaling,
Self-similarity
?-3
??
log u at R
?-cube model, ?-3 Similarity condition Mo ??
L3 ?o ?? L-1 ???? ?? ?o-3
log freq. (?)
This defines a scaling law. Spectral curves
differ by a constant factor at a given period
(e.g., 20 s), but they have the same high-freq.
asymptote
This behavior is expected when the nucleation
phase is responsible for the high-freq. asymptote
--but consider problem of time domain implication
for amplitude (Mb decreases with Mo)
4
Earthquake Source Properties, Spectra, Scaling,
Self-similarity
?-2
??
log u at R
?-square model, ?-2 Two possible
explanations 1) !Similarity condition
(not-similarity) Mo ?? L2 ?o ?? L-1 ???? ??
?o-2 2) Have similarity condition in terms of
nucleation, but high-freq. asymptote is produce
by stopping phase if rupture stops very abruptly
log freq. (?)
5
Earthquake Source Properties, Spectra, Scaling,
Self-similarity
Displacement and acceleration source spectra.
Spectra zero-frequency intercept (Mo), corner
frequency (?o or fc), high frequency decay (???),
maximum (observed, emitted) frequency fmax
?-n
log u at R
??
log freq. (?)
Aki, Scaling law of seismic spectrum, JGR, 72,
1217-1231, 1967. Hanks, b Values and ?-? seismic
source models implications for tectonic stress
variations along active crustal fault zones and
the estimation of high-frequency strong ground
motion, JGR, 84, 2235-2242, 1979.
6
Earthquake Source Properties, Spectra, Scaling,
Self-similarity
Source spectra for two events of equal stress
drop omega cube model
Large and Small Eq
L
?-3
S
log u at R
log freq. (?)
High-freq. spectral properties produced by
rupture growth, represent nucleation and
enlargement
7
Source spectra for two events of equal stress
drop omega square model
Large and Small Eq
L
?-2
S
log u at R
log freq. (?)
High-freq. spectral properties produced by
rupture growth, represent nucleation and
enlargement
8
Earthquake Source Properties, Spectra, Scaling,
Self-similarity
Relation between source (a) displacement (b)
velocity (c) acceleration history and asymptotic
behavior of spectrum
9
Earthquake Source Properties, Spectra, Scaling,
Self-similarity
Hanks (1979) 0. Consider two events that differ
in size by 10x and assume self-similarity of
rupture so that their moments differ by a factor
of 103 and durations differ by a factor of 10
(rupture velocity is the same and constant for
each event.) Take the events to be large enough
so that their corner frequency is well below 1
sec. 1. Four cases for time-domain
interpretation of spectral source models, 2 for
each model. ?-square (spectral amplitude at 1 sec
period is 10x greater for larger event) a) If
1-s. energy arrives continuously over the
complete faulting duration. Then 1-s time
domain amplitude is the same and Mb is the same
for both events Mb is independent of Mo. b) If
all 1-s energy radiated at the same time
(arrives at the same time) Then 1-s time domain
amplitude is 10x larger for the larger event Mb
scales directly with Mo. ?-cube (spectral
amplitude at T1 sec is the same for each
event) c) If 1-s energy arrives continuously
over the complete faulting duration. Then 1-s
time domain amplitude is smaller for the larger
event Mb scales inversely with Mo. d) If all
1-s energy radiated at the same time (arrives at
the same time) Then 1-s time domain amplitude
is the same for both events Mb is independent of
Mo.
10
Earthquake Source Properties, Spectra, Scaling,
Self-similarity
Hanks (1979) 0. Consider two events that differ
in size by 10x and assume self-similarity of
rupture so that their moments differ by a factor
of 103 and durations differ by a factor of 10
(rupture velocity is the same and constant for
each event.) Take the events to be large enough
so that their corner frequency is well below 1
sec. 1. Four cases for time-domain
interpretation of spectral source models, 2 for
each model. ?-square (spectral amplitude at 1 sec
period is 10x greater for larger event) a) If
1-s. energy arrives continuously over the
complete faulting duration. Then 1-s time
domain amplitude is the same and Mb is the same
for both events Mb is independent of Mo. b) If
all 1-s energy radiated at the same time
(arrives at the same time) Then 1-s time domain
amplitude is 10x larger for the larger event Mb
scales directly with Mo. ?-cube (spectral
amplitude at T1 sec is the same for each
event) c) If 1-s energy arrives continuously
over the complete faulting duration. Then 1-s
time domain amplitude is smaller for the larger
event Mb scales inversely with Mo. d) If all
1-s energy radiated at the same time (arrives at
the same time) Then 1-s time domain amplitude
is the same for both events Mb is independent of
Mo.