Physical interpretation
of DC and non-DC components
of moment tensors
Václav Vavryčuk
Institute of Geophysics, Prague
MT for simple types
of seismic sources
Explosive (implosive) source
Source process
Force equivalent
Moment tensor
M1
M 0

 0
three linear
dipoles
0
M1
0
0
0

M 1 
T rM  0 explosion
T rM  0 implosion
Shear faulting
Source process
Force equivalent
Moment tensor
 0 0 M0 
M   0 0 0 
M 0 0 0 
fault
no torque
double-couple
(quadrupole)
Pure tensile faulting
Source process
Force equivalent
faultfault
fault opening
Moment tensor
M1
M 0

 0
0
M1
0
0 
0 

M 2 
T rM  0 opening
T rM  0 closing
Shear-tensile faulting
Source process
Force equivalent
DC +
fault fault
fault opening
Moment tensor
 M1
M 0

 M 0
0
M1
0
M0
0 

M 2 
T rM  0 opening
T rM  0 closing
Decomposition of MT
Decomposition of MT
M  M ISO  M DC  M CLVD
T rM 
M*  M 
+
Tr M 
I
3
+
ISO
DC
non-shear
shear
CLVD
non-shear
Double-couple component of the moment tensor
Force equivalent
DC part
(shear faulting)
Seismic moment tensor
 0
M   0
 M 0
0 M0
0 0 
0 0 
double couple (DC)
 M 0
M   0
 0
0 
0 0 
0 M 0 
0
rotated double-couple (DC)
Non-double couple components: ISO and CLVD
Force equivalent
ISO part
(explosion)
CLVD part
(tensile crack)
Seismic moment tensor
M 1
M   0
 0
 M 2
M   0
 0
0
M1
0
0
 M2
0
compensated linear vector dipole
0 
0 
M 1 
0 
0 
2 M 2 
Decomposition of the moment tensor
M  M ISO  MCLVD  M DC
M
– moment tensor
MISO – trace of M
M*
M*
– deviatoric part of M
Percentage of ISO, CLVD and DC : |ISO|+|CLVD|+DC = 100%
ISO 
1 T rM 
 100%
3 M
MAX
DC  100%  ISO  CLVD
CLVD   2
M *MIN
M
*
MAX
100%  ISO 
Physical interpretation of MT
DC components
Parameters of shear faulting
• orientation of active faults, fault mapping
• type of fracturing (strike-slip, normal/reverse faulting)
Determination of present-day tectonic stress
• orientation of principal stress axes
• ratio between principal stresses
Numerical errors of the MT inversion
•
•
•
•
insufficient number of stations
presence of noise in the data
unfavourable station coverage of the focal sphere
inaccurate knowledge of the structure model
• approximate location and Green’s functions
• approximate moment tensor
Presence of single forces
no dipole forces
surface
Examples: impact of meteorites, landslides, volcanic eruptions, fluid
flow in volcanic channels
the process is not described by the moment tensor!
Complex shear faulting
DC1
DC2
DC + CLVD
fault
ISO = 0
Sum of two DCs of different orientations produces DC and CLVD
Combined shear-tensile faulting I


[u]
DC + CLVD + ISO


fluid pressure
fault
Example: hydrofracturing
High pore pressure can cause opening faults during the rupture process
(CLVD and ISO are then positive).
Combined shear-tensile faulting II
100
100
  0.2
  1.0
DC, CLVD, ISO [%]
80
80
DC
ISO
60
40
40
ISO
20
20
0
0
30
 
60
CLVD and ISO are correlated!
CLVD
60
CLVD
0
DC
90
0
30
 
60
ISO/CLVD -> vP/vS
90
Correlation between ISO and CLVD
ISO [%]
different vP/vS ratios
shear-tensile faulting
linear dependence
CLVD [%]
Shear faulting in anisotropic media
DC + CLVD + ISO
n
u
S
M kl
fault
 M 11
  M 12

 M 13
M 12
M 22
M 23
M 13 
M 23 

M 33 
The relation between fracture geometry and acting forces is more
complicated in anisotropic media than in isotropic media.
Moment tensors in isotropy
Shear earthquakes in isotropy
n
(Aki & Richards 2002, Eq. 3.22):
M kl  uS k nl  l nk 
0 0 1
M kl  M 0 0 0 0


1 0 0
double-couple (DC) mechanism
u
– slip
– fault area
 – shear modulus
 – slip direction
n – fault normal
cijkl – elastic parameters
u
S
S
Moment tensors in anisotropy
Shear earthquakes in anisotropy
n
(Aki & Richards 2002, Eq. 3.19):
M kl  uScijkl k nl
 M 11
M kl   M 12

 M 13
M 12
M 22
M 23
u
M 13 
M 23 

M 33 
general (non-DC) mechanism
– slip
– fault area
 – shear modulus
 – slip direction
n – fault normal
cijkl – elastic parameters
u
S
S
Examples
1997 West-Bohemian
earthquake swarm
Example: seismicity in West Bohemia
• active tectonics
• geothermal area
• mineral springs
• emanations of CO2
earthquake swarms:
• 1985/86
• 1994
• 1997
• 2000
• 2008
Swarm 97: Basic characteristics
N
• Duration:
2 weeks
• Number of
earthquakes:
P wave
1800
• ZStrongest event:
M=3.0
• Depth of events:
8.5-9.5 km
• Focal area:
700x700x1000 m
S wave
E
• Faults activated:
1s
2 different faults
Swarm 97: epicentres and mechanisms
Nový Kostel focal area
Fischer & Horálek (2000)
Horálek et al. (2000)
DC & non-DC mechanisms
Type A
Type B
12
12
A events
10
B events
10
8
8
N
6
6
4
4
2
2
0
0
0
20
40
60
DC [%]
80
100
0
20
40
60
DC [%]
80
100
N
Correlation between ISO and CLVD
corr. coeff = 0.91
20
+
vP/vS = 1.48
ISO [%]
10
0
strong indication for
tensile faulting!
-10
-20
0
20
CLVD [%]
40
Shear & tensile faulting
A events
B events
Shear faulting
Tensile faulting
u

u


Slip is along the fault
Slip is not along the fault
Moment tensor is DC
Moment tensor is non-DC
(DC+CLVD+ISO)
 – fault , u – slip,  – deviation of the slip from the fault
Deviation of the slip from the fault
6
8
A events
B events
6
4
4
N
N
2
2
0
0
-15
-10
-5
0
5
10
15
0
5
10
15
20
25
 [º]
 [º]
A events: -5 <  < 5
B events: 10 <  <20
30
Induced microseismicity during
the 2000 fluid injection experiment
in KTB, Germany
KTB superdeep drilling hole
• Nlocation: northern Bavaria, Germany
• holes:
pilot hole - 4 km, main borehole - 9.1 km
(October
1994), distance
P wave
S wave – 185 m
• Zgeology: crystalline unit, steeply inclined layers of
gneisses, amphibolites
• borehole geometry: vertical to 7.5 km, then inclined
• Ebottom hole temperature: 265ºC
1s
Injection experiment 2000
N
• experiment:
60 days
• amount of fluid: 4000 m3 of fresh water
P wave
wave
• entire 9.1 km
borehole wasSpressurized
Z
• well
head pressure was between 20 to 30 MPa
• flow rate ranged between 30 to 70 l/min
• several sharp pressure drops during shut-in phases
E
1s
Focal mechanisms of 37 events
Nodal lines
P/T axes
P
T
Nodal lines and P/T axes are well clustered
Non-DC components
CLVD percentage
ISO percentage
6
Number of earthquakes
Number of earthquakes
6
4
2
0
-40
-20
0
20
ISO [%]
Mean value of ISO = 1.5%
40
4
2
0
-80
-60
-40
-20
0
20
40
60
80
CLVD [%]
Mean value of CLVD = -5.7%
Positive values - tensile components, negative values - compressive components
Correlation between ISO and CLVD
corr. coeff. = 0.01
40
ISO [%]
20
no correlation!
0
-20
strong indication for
other origins than
tensile faulting!
-40
-80
-40
0
CLVD [%]
40
80
Anisotropy models at KTB
S-wave velocity
P-wave velocity
P wave
S waves
4.2
phase velocity [km/s]
phase velocity [km/s]
6.4
5.6
4.8
3.8
3.4
3
0
30
60
angle [degrees]
90
0
30
60
90
angle [degrees]
P anisotropy: 2 - 18%, S1 anisotropy: 3 - 18%, S2 anisotropy: 8 - 26%
anisotropic models of gneiss inferred from: VSP, sonic logs, lab measurements
References: Jahns et al. (1996), Rabbel (1994), Rabbel et al. (2004)
Inversion for anisotropy: method
Input:
• moment tensors of 37 events
• anisotropy at the focal area
Output:
• fault normals and slip directions
• theoretical DC, CLVD and ISO components
The misfit function: misfit between the
theoretical and observed non-DC components
Result: optimum orientation of anisotropy
Inversion for anisotropy: results
Misfit for ORT
Misfit for TI
a
b
o
o
o
o
The misfit function is normalized so it equals 1 for an isotropic medium.
Optimum anisotropy orientation
Triangles:
optimum orientation from
moment tensors
Squares:
orientation from MSP
Rabbel et al. (2004)
Anisotropy axes (plunge/azimuth):
Axis 1: 5º/65º, Axis 2: 50º/160º, Axis 3: 40º/330º
Summary
Significance of the DC components
Parameters of shear faulting
• orientation of active faults, fault mapping
• type of fracturing (strike-slip, normal/reverse faulting)
Determination of present-day tectonic stress
• orientation of principal stress axes
• ratio between principal stresses
Significance of the non-DC components
Discrimination explosions versus earthquakes
Analysis of tensile faulting
• detection of overpressure regime
• temporal variation of pore pressure
• the vP/vS ratio in a focal area
Estimation of anisotropy in a focal area
• orientation of anisotropy axes
• anisotropy strength