Earthquake source
modelling by second
degree moment tensors
Petra Adamová
Jan Šílený
Geophysical Institute, Academy of Sciences, Prague, Czech Republic
e-mail: adamova@ig.cas.cz, fax: +420-272761549
Introduction, motivation
• Finite source parameters from point source approximation
• traditional modeling of slip on
fault plane is more complicated
• 2nd degree moments are
adventageous alternative
 size of the source, duration of
the source process, average slip on
the fault, etc.
Theory: second degree moment tensors
First degree moment tensor representation:
Second degree moment tensor representation (Taylor expansion up to
degree two):
Second degree moments, Doornbos
(1982)
Standard MT
1. Time derivative of the response function (1 parameter): temporal
centroid – origin time of the finite extent source estimate
2. Spatial derivative (3 parameters): spatial centroid position
3. Combination of temporal and spatial derivative (3 parameters)
4. Second time derivative (1 parameter): source duration
From 3 and 4: rupture propagation along the fault
5. Second spatial derivative (6 parameters): geometrical
characteristics of the source (source ellipsoid)
Application for better estimate of
mechanism
• High non-DC component is reported for some strong events
by seismological agencies (Harvard, USGS, SED)
• This component is often questionable (large events, tectonic
origin)  it can be false due to unmodeled source finiteness
(strong event is modeled as point source)
• the scalar moment underestimation in the agency solution
 we will try to verify this hypothesis using synthetic test
Example of high non-DC
component
Izmit earthquake: agency solution (ETH)
Date/Time: 99/ 8/17 0: 1:38
Latitude 40.640
Longitude 29.830
Mw= 7.52
Strike = 90
Dip = 72
T
P
N
Rake = -164
DC = 59 %
CLVD = 41 %
ISO = 0 %
Very high non-DC component
Synthetic test: configuration
• Green’s functions are computed by DWN method
• crustal model is identical for data and synthetics (Bulut et al., 2007)
• noise-free data
Rupture model (J. Burjánek)
Unilateral rupture
Fault size: 20 km x 10 km
Scalar seismic moment: 1e18 Nm
f = 0 - 2 Hz
Rupture velocity 2.8 km/sec
Inversion scheme
Additional constraint: the volume of the focus is non-negative (McGuire et al.,
2001, 2002)
Synthetic data
unfiltered synthetic data
demonstrating the source
directivity:
station SDL: direction
perpendicular to the fault
strike.
station HER: ‘reverse’
direction
station BAL: ‘forward’
direction
Results: exact data
Common MT, f = 0.02 - 0.08 Hz
(3rd order Butterworth filter)
Theoretical
mechanism
P
T
N
Strike = 93
Dip = 73
Rake = -178
DC = 78 %
CLVD = 12 %
V = 10 %
Strike = 90
Dip = 72
Rake = 180
DC = 100 %
CLVD = 0 %
V=0%
Frequency test
• low-pass filtering as much as possible:
low-pass 3rd order Butterworth filter with a low-cut off at 0.02 Hz
• high-pass filter as much as possible but to keep the 2nd
degree effects
high-pass 3rd order Butterworth filter with a cut off at 0.1, 0.2, 0.3 and 0.4 Hz
Geometrical characteristics
Second spatial derivative, 6 parameters
A – 0.1 Hz
B – 0.2 Hz
C – 0.3 Hz
D – 0.4 Hz
frequencies used in the
inversion of 2nd degree
moments
Optimum frequency range is up
to 0.2 Hz
MT refinement: exclusion of
2nd degree terms
Refined MT: common MT without second degree terms
Strike =93
Dip = 73
Rake =2
DC = 78 %
CLVD = 12 %
V = 10 %
Theoretical
mechanism
P
T
N
T
P
N
Strike =93
Dip = 73
Rake =2
DC = 94 %
CLVD = 4%
V=2%
Strike = 90
Dip = 72
Rake = 180
DC = 100 %
CLVD = 0 %
V=0%
Reconstructed mechanisms
0.02-0.08 Hz
0.02- 0.1 Hz
0.02-0.08 Hz
0.02-0.3 Hz
0.02-0.08 Hz
0.02-0.08 Hz
0.02-0.2 Hz
0.02-0.4 Hz
Left: the mechanism obtained by inverting data filtered outside 0.02 -0.08 Hz
Right: mechanism from data corrected for the contribution of the 2nd degree
moments
frequency used in the inversion of 2nd degree moments
Test of robustness
Experiments simulating inconsistencies during the data inversion
a) source mislocation (1 km E, 1 km S, 2 km Z)
larger error in depth than in the horizontal coordinates simulates
smaller location precision
b) inaccurate GF (less layers + deviation 10% in each layer)
dashed line – simplified model
c) noise in data (15 - 30% from the maximal amplitude)
Geometrical characteristics
Second spatial
derivative, 6
parameters
Bold line – exact data
A - mislocation of the hypocenter when evaluating Green’s function
B - mismodeling of the velocity profile: the true 1-D model used to synthesize
the data, simplified when evaluating Green’s function
C - noisy data
Propagation vectors
exact data (black)
(a) hypocenter
mislocation
(b) the seismic
velocity profile
mismodeling
(c) noisy data
Background: vertical projection of the source model: the moment density
distribution of the unilaterally propagating rupture together with the 1 s, 2 s and 3 s
isochrones.
Reconstructed mechanisms
Left: the mechanism obtained by inverting data filtered outside 0.02 - 0.08 Hz
Right: mechanism from data corrected for the contribution of the 2nd degree
moments
Synthetic data vs. synthetic
seismograms
Black: synthetic data
Upper gray: synthetic
seismograms
Lower gray: 2nd degree
terms
station SDL: direction
perpendicular to the fault
strike.
station HER: ‘reverse’
direction
station BAL: ‘forward’
direction
Frequency range 0.02 -0.2 Hz
Conclusions
• We removed false non-DC component from the data
• Scalar seismic moment is higher with 2nd term than with
only 1st degree term
• Method of the second degree moments is perspective for
applications