Synthetic tests of slip inversion (two methods: old = ISOLA, new = conj. gradients)

J. Zahradník, F. Gallovič

MFF UK

Synthetic tests of slip inversion (two methods: old = ISOLA, young = conj.

gradients)

J. Zahradník, F. Gallovič

MFF UK

Motivation

New and old method of slip inversion:

how to understand the results for

a case study.

Movri Mountain (Andravida)

M6.3 earthquake, June 8, 2008

NW Peloponnese, GreeceCooperation: D. Křížová, V. Plicka, E. Sokos, A. Serpetsidaki, G-A. Tselentis

ITSAK, Greece

2 victimshundreds of injuries

HYPO and DD relocation: A. Serpetsidaki, Patras

PSLNET BB and SM (SER, MAM, LTK, PYL co-operated by Charles Univ.)

ITSAK SM NOA BB

A line source model

Gallovič et al., GRL, in press.

fixed foc. mech.:strike 30°,

dip 87°,rake -178°

Part 1

Synthetic tests

(error-free data only)

0 10000 20000 30000 40000fa u lt d istance (m )

0

4E+016

8E+016

1.2E+017

moment (Nm)

Model setup2 asperities along the fault ( azimuth 30°)rupture velocity: Vr = 3 km/s3 scenarios of rupture propagation: from the left, right and middle8 stations: as for the earthquakecrustal model: 1-D; Haslinger et al., 1999full-wave synthetics, 0.01-0.20 Hz

Goal: to find the slip evolution (x-t)without knowledge of hypocenter and Vr

Methods: Iterative method of F.G. (new) and ISOLA (old)Inversion makes use of the same Green function as in forward calculation.

0 10000 20000 30000 40000fa u lt d istance (m )

0

4E+016

8E+016

1.2E+017

moment (Nm)

0 10000 20000 30000 40000fa u lt d istance (m )

0

4E+016

8E+016

1.2E+017

moment (Nm)

x

mom

ent

Even date-time: 20040712 12:25:27Displacement (m). Inversion band (Hz) 0.01 0.03 0.1 0.2

ObservedSynthetic

Gray waveforms weren't used in inversion.

Blue numbers are variance reduction

-202

x 10-3 NS

MA

5 0.60

-10-505

x 10-3 EW

0.84

-202

x 10-3 Z

0.85

-505

10x 10

-3

SE

5 0.82

-10-505

x 10-3

0.80

-1012

x 10-3

0.79

-2-101

x 10-3

zak 0.90

-10123

x 10-3

0.87

-101

x 10-3

0.84

-101

x 10-3

ka

l 0.91

-101

x 10-3

0.94

-101

x 10-3

0.94

-202

x 10-3

LT

5 0.840

1020

x 10-4

0.88

-101

x 10-3

0.90

-101

x 10-3

PY

5 0.93-505

x 10-4

0.89

-101

x 10-3

0.93

-2-101

x 10-3

TH

L 0.80

-505

x 10-3

0.83

-101

x 10-3

0.80

-101

x 10-3

RG

A 0.80

-202

x 10-3

0.83

-505

x 10-4

0.83

0 50 100 150-4-2024

x 10-3

Time (sec)

XO

R 0.81

0 50 100 150-4-20246

x 10-3

Time (sec)

0.80

0 50 100 150-10-505

x 10-4

Time (sec)

0.77

Typical waveform fit for synthetic tests (varred ~ 0.9, both methods)

Iterative method

(F.G.)

x

t

position time cumul. mom. varred 31 28.85 .206330E+19 .7309E+00 10 22.55 .320968E+19 .9102E+00 34 33.05 .350033E+19 .9254E+00 29 23.15 .379023E+19 .9394E+00 38 20.15 .391736E+19 .9426E+00

Iterative method and

‘free’ ISOLA

x

t

20 subevents, cumulative moment 0.3e19, varred 0.99

Iterative method and

ISOLA ‘controlled’

The ‘control’ means a constraint imposed on the moment of each subevent. Instead of the automatically requested value Mo (sub i),only Mo/4 is adopted.

Vr = 3.28 km/s(instead of 3 km/s)

Unilateral propagation(from the left)

Vr = 3.68 km/s(instead of 3 km/s)

Unilateral propagation(from the right)

Vr = 5.68 and 5.26 km/s (instead of 3 km/s),

and a FALSE asperity in the middle

Bilateral propagation(from the center)

Partial results

The two methods give similar results, with similar problems.

The three scenarios behave in a different way (bilateral is the most problematic).

Part 2

How does it work ?

(A deeper insight into

the ‘correlation plots’).

ISOLA successively removes subevents following

the x-t ‘correlation’(max corr = likely slip x,t)

and so on ...

CORRELATION **2 = VARRED

SLIP HISTORY

Sub 1

Sub 2

Sub 3

Terminology:‘correlation’ plot,

‘correlation’ analysis,...

X

t

Plotted at an x-t position is the overall variance reduction (match between complete obs and syn waveforms at all stations);

the syn waveforms are calculated for a single point-sourceat that respective x-t node point, with moment adjustment.

Varred = c2,

where c is the correlation.

For simplicity, for c2, we often use term ‘correlation’.

X

t

Plotted at an x-t position is the overall variance reduction (match between complete obs and syn waveforms at all stations);

the syn waveforms are calculated for a single point-sourceat that respective x-t node point, with moment adjustment.

Varred = c2 ,

where c is the correlation

For simplicity, for c2, we often use term ‘correlation’.

t

waveform at a station

fittingwith Mo=1

adjustingMo

To understand the slip inversion we have to simulate the correlation plots, to reveal their dependence on :

• Slip distribution

• Station distribution

We assume that the correlation analysis works like a (multiple-signal) detector, similar to kinematic location. Once a signal is detected in a waveform, all equivalent x-t points

providing the same arrival are mapped.

Trade-off between source position and time

Station Y

True position of a point asperity Xa

Trial position of a point asperity X

Tr (Xa) + T(Xa,Y) = const = Tr (X) + T(X,Y)

Knowing the asperity position and time, Xa and Tr(Xa), we can calculate all equivalent positions X and times Tr (X) characterized by

the same arrival time (=const): a hyperbola. For a station along the source line, the Tr = Tr(X) degenerates to a straight line.

Simplified model of two asperities

(2 x 5 point sources)and

two stations0 10 20 30

a lo n g p ro file (km )

4

6

8

10

12

-100 -50 0 50 100Ea stin g (km )

-100

-50

0

50

100

No

rth

ing

(km

)

SE5

zak

0 10 20 30 40tim e (se c)

120

160

200

240

280

SE5

ZAK

Sergoula is directive.

Zakynthos is anti-directive station.

Seismograms contain 2 x 5 signals from the sources constituting the two asperities.

Matching complex seismograms with a single point source signals from the nodes of x-t grid (correlation analysis) we identify all possible

source positions.

Sergoula is directive.

Zakynthos is anti-directive station.

0 10 20 30 40tim e (se c)

120

160

200

240

280

320

SE5

ZAK

x

t

Signals from here are not supported by ZAK and SER.

0 10 20 30 40a lo n g p ro file (km )

0

4

8

12

16

time

(se

c)

Forward directivity at Sergoula. Two asperities seen as two narrow strips,each one composed of 5 lines (invisible).

0 10 20 30 40a lo n g p ro file (km )

0

4

8

12

16

20

time

(se

c)

SER

ZAK

0 10 20 30 40a lo n g p ro file (km )

0

4

8

12

16

time

(se

c)

Forward directivity at Sergoula. Two asperities seen as two narrow strips,each one composed of 5 lines (invisible).

Backward directivity at Zakynthos. Two asperities seen as two broad strips,

each one composed of 5 lines (5 point sources).

Intersection = ‘bright spots’ of the correlation

(narrow strips = large correlation)

0 10 20 30 40a lo n g p ro file (km )

0

4

8

12

16

20

time

(se

c)

SER

ZAK

Vr = 3 km/secThe ‘bright spots’

relate with asperities,

but a very cautious

interpretation is needed.

0 10 20 30 40a lo n g p ro file (km )

0

4

8

12

16

20tim

e (

sec)

SER

ZAK

Vr = 3 km/sec

3 stations

0 10 20 30 40a lo n g p ro file (km )

0

4

8

12

16

20

time

(se

c)

SER

ZAK

Vr = 3 km/sec

RGA

-100 -50 0 50 100Ea stin g (km )

-100

-50

0

50

100

No

rth

ing

(km

)

M A5

SE5

zak

kal

LT5

PY5

TH L

R G A XO R

Simulation explains the correlation plots.

This is what we expected (assuming that the correlation analysis works as a detector).

This is what we obtained from correlation analysis of synthetic seismograms at the two stations.

0 10 20 30 40a lo n g p ro file (km )

0

4

8

12

16

20

time

(se

c)

SER

ZAK

Vr = 3 km/sec

RGA

Partial results

The correlation diagrams can be explained in terms of (multiple-source) location. It validates the assumption that correlation analysis works like a signal detector.

‘Strips’ and ‘bright spots’ in the correlation plots are due to relative shifts between multiple sources, as seen by

stations (directivity).

Correlation plot is nothing but a mapping of the shifted signals back to fault. The map (after some deciphering)

can be translated into the slip history.

Now we know (for a given slip history) how the

stations contribute to the correlation plot.

What else ?

0 10 20 30 40a lo n g p ro file (km )

0

4

8

12

16

20

time

(se

c)

SER

ZAK

Vr = 3 km/sec

Opposite: It would be nice to learn about the station contributions from the correlation plots (without knowing the slip history).It might help to understand limitations of the slip inversion.

0 10 20 30 40a lo n g p ro file (km )

0

4

8

12

16

20

time

(se

c)

SER

ZAK

Vr = 3 km/sec

0 10 20 30 40tim e (se c)

120

160

200

240

280

SE5

ZAK

Stripof ZAK ?

Stripof SE5 ?

Indeed, seismograms enable reconstruction of the single-station

strips: SE5 ZAK

0 10 20 30 40a lo n g p ro file (km )

0

4

8

12

16

20

time

(se

c)

SER

ZAK

Vr = 3 km/sec

Compared to the simple forward model, we face complexities:e.g. ZAK second strip is weaker and non-uniform.

Effect of distance from the station(fitting well weaker arrival

decreases the overall varred).

Effect of focal mechanism(foc mech is well fitted only at correct x ).

Fitting a 2-signal record by a single signal

(in real case signals are convolved with Green function, thus more complicated)

fittingwith Mo=1

adjustingMo

Varred is HIGH becausethe strong pulse is fitted

Varred is LOW becausethe weak pulse is fitted

more distantasperity providesa weaker signal

Single station: ZAKEven date-time: 20040712 12:25:27Displacement (m). Inversion band (Hz) 0.01 0.03 0.1 0.2

ObservedSynthetic

Gray waveforms weren't used in inversion.

Blue numbers are variance reduction

-202

x 10-3 NS

MA

5 0.64

-10-505

x 10-3 EW

0.37

-202

x 10-3 Z

0.52

-505

10x 10

-3

SE

5 0.45

-10-505

x 10-3

0.50

-1012

x 10-3

0.49

-2-101

x 10-3

zak 0.19

-202

x 10-3

0.77

-101

x 10-3

0.39

-101

x 10-3

ka

l 0.58

-15-10-505

x 10-4

0.84

-101

x 10-3

0.58

-202

x 10-3

LT

5 0.260

1020

x 10-4

0.51

-101

x 10-3

0.52

-101

x 10-3

PY

5 0.67

-505

x 10-4

0.13

-101

x 10-3

0.64

-2-101

x 10-3

TH

L 0.50

-505

x 10-3

0.52

-101

x 10-3

0.47

-101

x 10-3

RG

A 0.12

-202

x 10-3

0.33

-505

x 10-4

0.49

0 50 100 150-4-2024

x 10-3

Time (sec)

XO

R 0.48

0 50 100 150-4-20246

x 10-3

Time (sec)

0.48

0 50 100 150-10-505

x 10-4

Time (sec)

0.43

10 23.00 .963153E+18 .5366E+00

fitting closer asperity

Single station: ZAK

33 29.00 .666635E+18 .1894E+00

Even date-time: 20040712 12:25:27Displacement (m). Inversion band (Hz) 0.01 0.03 0.1 0.2

ObservedSynthetic

Gray waveforms weren't used in inversion.

Blue numbers are variance reduction

-202

x 10-3 NS

MA

5 -0.41

-10-505

x 10-3 EW

0.50

-202

x 10-3 Z

0.36

-505

10x 10

-3

SE

5 0.42

-10-505

x 10-3

0.35

-1012

x 10-3

0.32

-2-101

x 10-3

zak 0.63

-10123

x 10-3

-0.07

-101

x 10-3

0.24

-101

x 10-3

ka

l 0.35

-15-10-505

x 10-4

0.06

-101

x 10-3

0.38

-202

x 10-3

LT

5 0.570

1020

x 10-4

0.38

-101

x 10-3

0.42

-101

x 10-3

PY

5 0.34

-505

x 10-4

0.66

-101

x 10-3

0.32

-2-101

x 10-3

TH

L 0.34

-505

x 10-3

0.38

-101

x 10-3

0.37

-101

x 10-3

RG

A 0.44

-202

x 10-3

0.41

-505

x 10-4

0.19

0 50 100 150-4-2024

x 10-3

Time (sec)

XO

R 0.39

0 50 100 150-4-20246

x 10-3

Time (sec)

0.37

0 50 100 150-10-505

x 10-4

Time (sec)

0.36

fitting distant asperity

Single station: ZAKEven date-time: 20040712 12:25:27Displacement (m). Inversion band (Hz) 0.01 0.03 0.1 0.2

ObservedSynthetic

Gray waveforms weren't used in inversion.

Blue numbers are variance reduction

-202

x 10-3 NS

MA

5 0.64

-10-505

x 10-3 EW

0.37

-202

x 10-3 Z

0.52

-505

10x 10

-3

SE

5 0.45

-10-505

x 10-3

0.50

-1012

x 10-3

0.49

-2-101

x 10-3

zak 0.19

-202

x 10-3

0.77

-101

x 10-3

0.39

-101

x 10-3

ka

l 0.58

-15-10-505

x 10-4

0.84

-101

x 10-3

0.58

-202

x 10-3

LT

5 0.260

1020

x 10-4

0.51

-101

x 10-3

0.52

-101

x 10-3

PY

5 0.67

-505

x 10-4

0.13

-101

x 10-3

0.64

-2-101

x 10-3

TH

L 0.50

-505

x 10-3

0.52

-101

x 10-3

0.47

-101

x 10-3

RG

A 0.12

-202

x 10-3

0.33

-505

x 10-4

0.49

0 50 100 150-4-2024

x 10-3

Time (sec)

XO

R 0.48

0 50 100 150-4-20246

x 10-3

Time (sec)

0.48

0 50 100 150-10-505

x 10-4

Time (sec)

0.43

10 23.00 .963153E+18 .5366E+00 33 29.00 .185847E+19 .8781E+00

Rupture propagation from the right

Sum of 3

SE5

ZAK

RGA

corr01

Rupture propagation from the right

Sum of 3

corr01-40 -30 -20 -10 0

a lo n g p ro file (km )

0

5

10

15

20

25

time

(se

c)

SER

ZAK

Vr = 3 km/secRGA

Rupture propagation from the right

Sum of 3

corr01-40 -30 -20 -10 0

a lo n g p ro file (km )

0

5

10

15

20

25

time

(se

c)

SER

ZAK

Vr = 3 km/secRGA

= FALSE (making Vr estimate wrong, 3.68 instead of 3.00 km/s)

Rupture propagation from the center (bilateral)

corr01

SE5

ZAK

RGA

Rupture propagation from the center (bilateral)

corr01

-20 -10 0 10 20a lo n g p ro file (km )

0

4

8

12

16

time

(se

c)

SER

ZAK

Vr = 3 km/sec

SER is directive for the RIGHT asperity, ZAK for the LEFT one. And vice versa.

Crossing directive strips creates false asperity.Other stations (not crossing it) do not help.

Narrow ZAK strip has apparent velocity ~ 5 km/s,hence wrong estimate of Vr to the left

Similarly, due to RGA to the right (previous slide).

Rupture propagation from the center (bilateral)

corr01

-20 -10 0 10 20a lo n g p ro file (km )

0

4

8

12

16

time

(se

c)

SER

ZAK

Vr = 3 km/sec

RGA

= false

Partial resultsWe understood problems of the synthetic slip inversion: correct position and time of asperities can be only found when detecting

x-t regions crossed by strips from all stations. However, false bright spots are created by directive strips also in places not

crossed by other station strips.

The problem is most severe in case of several asperities, and mainly for the bilateral ruptures.

Surprisingly, although the explanation came from analysis of ISOLA method, it holds for the iterative method, too; the two

methods are more similar than expected.

Partial results - continuedDirective/antidirective stations are not just along the fault, but in a relatively wide angle (plus minus 30 deg) with respect to fault.

The mentioned problematic effects of directive/antidirective stations do not imply that these should be excluded from

inversion. Just the opposite is true ! They play a key role, but …

We believe that most problems of slip inversion is not in ‘poor station coverage’. Note, for example, that two stations at an

angle with respect to fault strike plus/minus are equivalent. Sometimes, perhaps, less stations would do a better job

(we mean good stations ).

Part 3 - outlookHow to avoid false asperities ?

A) Always perform detection analysis (find the correlation strips) from each station separately,

and guess if they might possibly create

false bright spots.

B) False intersections can be reduced in ISOLA by starting iterative deconvolution with a finite

source, well describing a true asperity.

In fact, the primitive ‘free’ ISOLA was an extreme case of such a method. The ‘controlled’ ISOLA

was worse:

In fact, the primitive ‘free’ ISOLA was an extreme case of such a method. The ‘controlled’ ISOLA

was worse:

And this is the best: the ‘controlled’ ISOLAwhose subevent 1 is a finite right-hand asperity:

Any implication for Andravida ?

A) We inspected contributions of all stations, found a very clear directivity,

but did not see any obvious false asperity.

MAM ZAK

0 50 100 150 200 250 300 350 400 450-0.01

-0.005

0

0.005

0.01

Dis

plac

emen

t (m

)

MA5

NS

RealSynthetic

0 50 100 150 200 250 300 350 400 450-0.02

-0.01

0

0.01

Dis

plac

emen

t (m

)

EW

RealSynthetic

0 50 100 150 200 250 300 350 400 450-0.01

-0.005

0

0.005

0.01

Dis

plac

emen

t (m

)

Time (Sec)

Z

RealSynthetic

0 50 100 150 200 250 300 350 400 450-0.02

-0.01

0

0.01

Dis

plac

emen

t (m

)

zak

NS

RealSynthetic

0 50 100 150 200 250 300 350 400 450-0.01

-0.005

0

0.005

0.01

Dis

plac

emen

t (m

)

EW

RealSynthetic

0 50 100 150 200 250 300 350 400 450-10

-5

0

5x 10

-3

Dis

plac

emen

t (m

)

Time (Sec)

Z

RealSynthetic

f < 0.2 Hz

MAM ZAK

0 50 100 150 200 250 300 350 400 450-0.01

-0.005

0

0.005

0.01

Dis

plac

emen

t (m

)

MA5

NS

RealSynthetic

0 50 100 150 200 250 300 350 400 450-0.02

-0.01

0

0.01

Dis

plac

emen

t (m

)

EW

RealSynthetic

0 50 100 150 200 250 300 350 400 450-0.01

-0.005

0

0.005

0.01

Dis

plac

emen

t (m

)

Time (Sec)

Z

RealSynthetic

0 50 100 150 200 250 300 350 400 450-0.02

-0.01

0

0.01

Dis

plac

emen

t (m

)

zak

NS

RealSynthetic

0 50 100 150 200 250 300 350 400 450-0.01

-0.005

0

0.005

0.01

Dis

plac

emen

t (m

)

EW

RealSynthetic

0 50 100 150 200 250 300 350 400 450-10

-5

0

5x 10

-3

Dis

plac

emen

t (m

)

Time (Sec)

Z

RealSynthetic

f < 1 Hz

all stations, f < 0.2 Hz

all stations; f < 1.0 Hz

Andravidacorrelation plots

Any other implication for Andravida ?B) We try ISOLA with different finite sub1, for example:

without a finite sub1

with a finite sub1

Even date-time: 20040712 12:25:27Displacement (m). Inversion band (Hz) 0.01 0.03 0.1 0.2

ObservedSynthetic

Gray waveforms weren't used in inversion.

Blue numbers are variance reduction

-505

10x 10

-3 NS

va

r 0.72

-505

10x 10

-3 EW

0.90

-4-202

x 10-3 Z

0.81

-4-202

x 10-3

MA

5 0.73

-8-6-4-202

x 10-3

0.97

-101

x 10-3

-0.10

-505

x 10-3

SE

5 0.95

-8-6-4-202

x 10-3

0.85

-202

x 10-3

0.06

-4-2024

x 10-3

zak 0.57

-2024

x 10-3

0.76

-1012

x 10-3

0.52

-1012

x 10-3

ka

l 0.63

-202

x 10-3

0.51

-10-505

x 10-4

-0.03

-4-202

x 10-3

LT

5 0.63

-1012

x 10-3

0.70-101

x 10-3

0.55

-101

x 10-3

PY

5 0.75

-202

x 10-3

0.27

-10-505

x 10-4

-0.06

-101

x 10-3

TH

L 0.30

-4-20246

x 10-3

0.78

-505

x 10-4

-1.52

-202

x 10-3

RG

A 0.49

-6-4-2024

x 10-3

0.72-101

x 10-3

-0.51

0 50 100 150-202

x 10-3

Time (sec)

XO

R -0.13

0 50 100 150-202

x 10-3

Time (sec)

0.10

0 50 100 150-101

x 10-3

Time (sec)

0.21

ConclusionsIn fact, we make location of (multiple) sources, on a given line, using waveforms,

i.e. without any phase picking. It is possible thanks to the fact that correlation analysis works as a signal detector. It is based on fitting complex waveform with a single pulse (s*G) from the node of x-t grid. This principle is implicitly common to both methods.

Since misfit function is a sum of L2 norms of individual stations, we get the same correlation plot from all stations processed simultaneously, or each station separately.

Separate analysis of individual stations is recommended. Due to directivity, each

station has different shifts between source signals (a high ‘signal density’ for directive stations). The signal density is mapped from the data space to the model space, where it produces the strips.

Intersections of the strips (mainly narrow/dense and broad/coarse for directive and

antidirective stations, resp.) are the correlation bright spots. Some of them are false. These have to be avoided prior interpreting bright spots in terms of slip.

Artifacts of slip inversion can be partially avoided when first removing the finite-

source effect of one important (true) asperity from the data.

Thank you for your attention.

Sorry if we confused you.Last month this exciting

(but a bit tricky) topic confused us 10 times every day …

Resolving two sources 7 km of each otherhorizontally and vertically

f < 1 Hz

f < 0.2 Hz

Station distribution: as for the Andravida earthquake

-100 -50 0 50 100Ea stin g (km )

-100

-50

0

50

100N

ort

hin

g (

km)

M A5

SE5

zak

kal

LT5

PY5

TH L

R G A XO R