2167-28
Advanced School on Direct and Inverse Problems of Seismology
27 September - 8 October, 2010
Relative location, relative moment tensor inversion
Torsten Dahm
Institut fuer Geophysik
Universitaat Hamburg
Germany
Relative location, relative moment
tensor inversion
ICTP Course 2010 Trieste
Torsten Dahm
[email protected]
Institut für Geophysik, Universität Hamburg
contributions from: Th. Fischer, J. Reinhardt
Dahm, ICTP 2010 III – p.1
Earthquake swarms & cluster
.0
45
.9
)
39
km
44
no 7
e 0.
a
st 2
(
.
rth
(k
4
.4
44
)
m
.5
40
.1
45
.0
44
no 4
rth
40
.
.2
44
(k
m
ea
s
-9.0
-9.3
.9
.7
39
height (km)
t (k40.2
.
44
44
-8.7
40
no 4
rth
)
m
(k
-9.0
.0
44
.1
45
40
44
m
(k
40
.2
.
(km
44
st
no 7
rth
ea
)
.4
.5
-9.3
39
)
.9
45
.0
height (km)
.1
44
44
.5
-8.7
40
.1
)
.5
st
m)
ea
(k
m
44
.9
)
.7
39
45
.0
)
44(
.k7m
no
rt
h
44
.4
44
.1
39.9
40.5
east40.2
(km)
45.0
40.2
(km)
40.5
east
39.9
44.7 north 44.4
(km)
44.1
700 clustered events from the 1997 Bohemian massive
earthquake swarm (see Dahm, Sileny and Horalek, 2000).
Dahm, ICTP 2010 III – p.2
A. Precise earthquake location
The simultaneous location of clustered earthquakes and
multiplets increases the location accuracy and reduces
model-dependent bias.
Dahm, ICTP 2010 III – p.3
Single source location (Geigers method)
= (X, Y, Z):
Equation for one observation at X
velocity)
t = tobs = h + T (x, X,
(summ. conv.)
linearizing around starting model m0
tobs − ttheo (m0)
≈
∂t
∂t
(m0k − mk ) =
Δmk
∂mk
∂mk
t
h
T
x = (x, y, z)
m
= (h, X − x, Y − y, Z − z)
with
:
:
:
:
:
arrival time
origin time
travel time
location vector
model vector
Dahm, ICTP 2010 III – p.4
master event location
station
Δ t = Δ t0 + Δ x / c
1
2
sources
far-field equation (i=slave event, j =master event):
n
obs
obs
ti
− tj
= (hi − hj ) + (xi − xj )
c
with
n : unit vector of ray to X
c : velocity at source
Dahm, ICTP 2010 III – p.5
master event location
relative locations with high accuracy, but absolute
location is not improved
location of master event is not changed
high precision time-difference measurements when
waveforms are similar
a minor influence of the unmodelled unknown structure
far away from the cluster
systematic time errors at a station have no effect on the
results
Dahm, ICTP 2010 III – p.6
Double difference method
Dahm, ICTP 2010 III – p.7
Double difference method (linearized)
The parallel ray assumption is relaxed. For each
:
observation at X
obs ) − (ttheo − ttheo ) =
(tobs
−
t
(i)
(j)
(i)
(j)
=
(j)
∂t(i)
∂t
(i)
(j)
Δmk −
Δmk
∂mk
∂mk
advantage
: bended rays, mixed input data
disadvantage : nonlinear, iterative scheme
accurate theoretical traveltime differences
(see Waldhauser and Ellsworth, BSSA, 2000)
Dahm, ICTP 2010 III – p.8
Double difference method
Relocation of 28 events from the Berkeley cluster
(Waldhauser and Ellsworth, BSSA, 2000)
Dahm, ICTP 2010 III – p.9
Application to 1997 Vogtland swarm
0.0
0.5
1.0
1.5
2.0
0.0
0.5
1.0
1.5
2.0
0.0
0.5
1.0
1.5
2.0
WEBNET
waveform data
available (Horálek,
pers. comm.)
Precise estimate of
time shift for best
coherence
Arrival time
differences with
high accuracy
Easy amplitude
picking
Dahm, ICTP 2010 III – p.10
A. accurate time difference measurment
0.0
0.5
1.0
1.5
2.0
very accurate arrival time dif0.0
0.5
1.0
1.5
2.0
0.0
0.5
1.0
1.5
2.0
ference measurements when waveforms are similar (correlation approach)
Dahm, ICTP 2010 III – p.11
B. multiplet analysis
Coherence analysis (Maurer and Deichmann, 1995)
with events of the 1997
earthquake swarm beneath
Novy Kostel
600
400
37 % of the events
(274) grouped in 14
multiplets
200
0
0
200
400
600
relative time differences
extracted
correlation
calculated
coefficients
Dahm, ICTP 2010 III – p.12
C. high precision relocation
002
016
115
031
481
327
225
319
287
352
431
494
517
697
500
0
20
-500
-500
0
500
15
y [m] (W->E)
1
2
3
4
5
6
7
8
9
multiplet No.
10
11
12
13
14
time [days]
x [m] (S->N)
25
1 2 3 4 5 6 7 8 9 10 11 12 13 14
multiplet no.
Dahm, ICTP 2010 III – p.13
D. Precise source mechanisms
For clustered earthquakes and for multiplets a relative amplitude inversion (relative moment tensor)
is possible and allows to analyse weak events.
Having accurate locations and source mechanism
is useful for the identification of micro-faults and
stress inversion
Dahm, ICTP 2010 III – p.14
relative moment tensor inverson
(i)
(i)
h
m
u
= k
k I
station
source
(1)
mk
I Green function
(2)
mk
(k=1,6)
Body waves amplitudes depend on a single scalar Green
function, that can be eliminated when one master event
mechanism is known
Dahm, ICTP 2010 III – p.15
relative moment tensor inversion
moment tensors relative to the tensor of a master event,
but absolute tensors (moments) are not improved
moment tensor of the master event is not improved
relative amplitudes are measured with high precision
when waveforms are similar
unmodelled unknown structure far away from the cluster
has a minor influence on the results
systematic station effects have no impact on the results
Dahm, ICTP 2010 III – p.16
assumptions
the mechanism of the reference event is well known a
priori
events are narrow clustered and seismograms are
lowpass filtered (temporal and spatial point source, all
events occur within approx. one wavelength from the
master event)
used frequency range below the corner frequency of
the largest studied event of the cluster
isolated, non-interfering body-waves are used
Dahm, ICTP 2010 III – p.17
Estimation of Moment Tensors
1
2
3
4
5
6
7
8
multiplet no.
9
10
11
12
13
14
Relative moment tensor inversion (Dahm, 1996)
automated picking of P- and S-phase amplitudes
522 moment tensors inverted
Classification with multiplets possible
Dahm, ICTP 2010 III – p.18
Plane with en echelon faults
z [m]
-500
0
x’ [m] (45° N)
500 -500
8500
500
8500
9000
9000
9500
9500
287 352 697 431 517
0
z [m]
y’ [m] (135° N)
287 352 697 431 517
1 2 3 4 5 6 7 8 9 1011 12 1314
multiplet no.
Dahm, ICTP 2010 III – p.19
body-wave approach
for each ray (P, SV, SH):
ũ = hk mk I
,
with
hP1 = − cos(2ϕ)sin2 α ,
hP2 = sin(2ϕ)sin2 α ,
hP3 = cos ϕsin2α ,
hP4 = sin ϕsin2α ,
hP5 = 2 − 3sin2 α ,
hP6 = 1 ,
and similar for SH-waves.
hSV
1
hSV
2
hSV
3
hSV
4
hSV
5
hSV
6
= −0.5 cos(2ϕ)sin2α ,
= 0.5 sin(2ϕ)sin2α ,
= cos ϕcos2α ,
= sin ϕcos2α ,
= −1.5sin2α ,
= 0,
(e.g. Aki & Richards, 1982)
Dahm, ICTP 2010 III – p.20
used convention
take-off angle: α
azimuth angle: ϕ
moment tensor:
m1 =
m2 =
m3 =
m4 =
m5 =
m6 =
1
(M22 − M11 )
2
M12
M13
M23
1
(0.5(M22 − M11 ) − M33 )
3
1
(M11 + M22 + M33 )
3
Dahm, ICTP 2010 III – p.21
what is different?
the ”Green function I ” is a scalar for each body-wave
mode
=⇒ it can be eliminated when using two earthquakes from
the same source region
(1)
ũ
=
I =
(1) (1)
hk mk I
ũ(2)
(2)
(2)
hk mk
leading to
(2)
ũ
(1) (1)
hl ml
(1)
= ũ
(2) (2)
hk mk
T
[data] = [vector ] [unknown model]
Dahm, ICTP 2010 III – p.22
additional polarity constraints
(1)
(1)
ũ(2) hl ml
(2)
(2)
= ũ(1) hk mk
ũ(2) (2) (2)
0 ≤
hk mk
(2)
|ũ |
[data] = [vector ]T [unknown model ]
0 ≤ polarity · [ geometry ]T [unknown model ]
Dahm, ICTP 2010 III – p.23
relative method without master
From
(1) (1)
hl ml
(2)
ũ
(1)
= ũ
(2) (2)
hk mk
we formally write
0
2
4
0
const
=
2
3
5
(1)
−ũ(2) hl
=
4
(1)
−ũ(2) h1
1
(1)
ml
(2)
(2)
+ ũ(1) hk mk
...
...
(2)
−ũ(2) h6
1
2
(2)
ũ(1) h1
1
...
...
(2)
ũ(1) h6
1
6
6
6
6
6
6
36
6
6
56
6
6
6
6
6
6
6
6
4
(1)
m1
(1)
m2
.
.
.
(1)
m6
(2)
m1
(2)
m2
.
.
.
(2)
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
m6
Dahm, ICTP 2010 III – p.24
Hindu Kush deep cluster
Hindu Kush
(a)
(b)
70˚ 30'
71˚ 00'
20 km
36.75
36˚ 45'
36˚ 30'
8 11
10
10
8
9
7
2
2
6
7
3
36˚ 15'
5
9
1
1
6
36.50
11
3
5
4
36.25
4
175
200
225
z[km]
250
Dahm, ICTP 2010 III – p.25
TAM
INU
SSB
ECH
YAK
ATD
BJT
OBN
ARU
KIV
KURK
BRVK
AAK
WUS
ABKT
NIL
record section for event No 1
t - x/8 (s)
400
300
sS
200
S
PcP
100
0
P
-100
PP
pP sP
-200
-300
1000
2000
3000
4000
5000
6000
7000
x (km)
Dahm, ICTP 2010 III – p.26
data processing
11 events with max. distance of 35 km.
band-pass filter between 0.05 Hz and 0.1 Hz;
wavelength between ≈ 80 km (P) and ≈ 46 km (S)
P-, S-, pP-, sP-, PP-phases selected where no
interference
peak amplitudes measured
Dahm, ICTP 2010 III – p.27
measuring amplitudes
Dahm, ICTP 2010 III – p.28
inversion results
Dahm, ICTP 2010 III – p.29
programs and directories
directory example
1. relref.exe: relative inversion with reference mechanism
2. relrpos.exe: relative inversion without reference
mechanism
3. syndat4relef.f: generate synthetic input to test geometry
4. pltbeachball.cmd: plot solutions in comparison to
Harvad CMT
Dahm, ICTP 2010 III – p.30
relref.f
Input: relref.inp
Output:
1. relref.out: full listing of result
2. result.par: short version of result for plotting
3. relray.pola: polarities used
4. . . .
Dahm, ICTP 2010 III – p.31
description of input file
HK, 11ev, 28.07.99, Pol
iev ibit inorm idyn iclust imat idev ipol
11
56
+1
0
0
0
6
0
fact(i),i=1,iev (16f5.1)
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
.00E+00
1.0
1.0
1.0
1.0
1.0
1
.0000
wt
cmp
azi
toff / u(i,j), i=1,iev
1.00
1
23.062
102.630
AAK P
0.000
0.000
0 -7.9522000000000e+03
25.644
98.997
0.0000 1.000e+00 -1
0 -1.9283100000000e+05
23.072
99.817
0.0000 1.000e+00 -1
1 0.0000000000000e+00
21.440
100.23 4
0.0000 1.000e+00 0
0 -7.2773000000000e+04
23.992
106.039
0.0000 1.000e+00 -1
......
Reference Mechanisms
1 0
-0.76
0.07
.....
.....
0.00
-0.63
0.12
0.76
0.40
Dahm, ICTP 2010 III – p.32
parameter description (formatted input!)
line 1
:
text, not used
line 3
:
iev: No of events
:
ibit: No of extracted phases per event (max)
:
inorm: 1 defines fact(i)=1
:
idyn: 0 defines weighting with wt(j) of each phase
:
iclust: 0 azimuth (ϕ) and take-off angle (α) defined only by event 1
:
iclust: 1 take-off and azi as given in phase reading line
:
idummy not used
:
idev 5 deviatoric moment tensor constraint
:
ipol 1 if polarity constraints should be used
line 5
:
(16f5.1) (fact(i),i=1,iev) to weight individual events (not recommended)
line 13 ff
:
event block-wise reading of phase amplitudes
last line(s)
:
definining moment tensor of reference event(s)
HK, 11ev, 28.07.99, Pol
iev ibit inorm idyn iclust imat idev ipol
11
56
+1
0
0
0
6
0
fact(i),i=1,iev (16f5.1)
1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0
1.0
1.0
1.0
1.0
1.0
Dahm, ICTP 2010 III – p.33
1
ibit blocks of phase amplitudes
head line
block of size
iev
:
wt: weight,
:
cmp: compenent (1=ver,2=rad,3=trans),
:
azi: mean azimuth,
:
toff: mean take-off,
:
cstat: station name,
:
cph: phase name, must begin with either P (P-wave) or S (S-wave)
:
isk(i,j): phase reading may be scipped (except polarity constraint)
:
u(i,j): measured amplitude
:
azi(i,j): azimuth of ray
:
toff(i,j): take-off angle of ray
:
rm: dummy value, not used
:
tdiff: relative arrival time difference (not used here)
:
f1: geometrical spreading correction factor (if iclust=2)
wt
cmp
azi
toff / u(i,j), i=1,iev
1.00
1
23.062
102.630
AAK P
0.000
0.000
0 -7.9522000000000e+03
25.644
98.997
0.0000 1.000e+00 -1
0 -1.9283100000000e+05
23.072
99.817
0.0000 1.000e+00 -1
1 0.0000000000000e+00
21.440
100.23 4
0.0000 1.000e+00 0
0 -7.2773000000000e+04
23.992
106.039
0.0000 1.000e+00
-1
Dahm, ICTP 2010 III – p.34
reference event(s) moment tensors
add reference event lines as needed, each containing
kref
:
No of reference event as given in the data-block
is
:
(0) give moment tensor as M11 , M12 , M22 , M13 , M23 , M33
xmref(1-6)
:
six moment tensor components as defined by is
rmref
:
scalar moment of reference event
Reference Mechanisms
1 0
-0.76
0.07
0.00
-0.63
0.12
0.76
0.40
Dahm, ICTP 2010 III – p.35
practical
1. reproduce results using relref.exe
2. introduce deviatoric source constraint
3. introduce polarity constraint
4. use only event 1 as reference event (relref.exe)
5. change moment tensor of reference event
1. edit input file (relref.inp) and make changes
2. run relref.exe
3. view output files, e.g. relref.out
4. plot results in comparison to CMT solutions
(e.g. pltbeachball.cmd)
Dahm, ICTP 2010 III – p.36