estimating uncertainty of moment tensor using singular vectors

Estimating uncertainty of moment tensor

using singular vectors.J. Zahradník

Charles University in PragueCzech Republic

Motivation

The cases exist that MT was successfully derived from a few stations.

When and why?

The volume component is highly unstable. How does it trade-off

with the other MT components?

How to solve these questions without seismograms?

Part 1

Theory(Numerical Recipes, chap. 2.6, 15.4, 15.6)

Linear problem, least squares,solution by means of normal equations.

Cautionr: data variance needed!

Linear problem, least squares,solution by means of SVD.

Advantage:

SVD expresses the uncertaintyby singular vectorsin a transparent way.

Cautionr: data variance needed!

Gaussian and non-gaussian errors

Non-linear problems and non-gaussian errors:

Ellipsoid substituted by an irregular (perhaps non-compact) error volume. The probability density function (pdf)has to be found experimentally.

!

Example what cannot be used if errors are non-gaussian.

In gaussian case the absolute size of the confidence interval can be found(related to a given probability level) if

we know the covariance matrix.

In practice, however, we often do not know data errors,

so the covariance still cannot be used

(even for gaussian errors) in absolute sense….

Example of error volumes of a non-linear problem.Probabilistic location (NonLinLoc, A. Lomax)

V. Plicka

Epicenter (2 parameters)

Depth (1 single parameter may have a specific physical meaning)

Highly important even in a relative sense

(without exact knowledge

of the data errors) !

1D pdf is a projection, not a section!

XX

Part 2Linear problem for MT

(position and source time fixed)Motivation: Greece -Dana Křížová

Portugal - Susana Custódio

Uncertainty in practice:10/5/2008

4 agencies may represent 4 solutions(for some events)

Instituto Andaluz de Geofísica (IAG)Instituto Geográfico Nacional (IGN)OBS stations (NEAREST)

Ana Lúcia das Neves Araújo da Silva Domingues(red beachballs)

ISOLA

ISOLA old:a) min/max eigenvalue ratio

of the matrix of normal equations

b) Variances of the parameters a1,…a6

ISOLA new:??????????????

Example to study the MT uncertainty

ISOLA new:

A) Perturbing the MT solution within the error ellipsoid

We need: crustal model, station positions, source position, and also the assumed MT.

We do not need neither the true MT, nor seismograms.

ISOLA produces the matrix, the singular vectors and values, and an auxiliary code generates the

ellipsoid.

6 stations, source depth 10 km

Presented is a 2D section, but we calculate a 6D elipsoid,

a1…a6.

The stardelta ai= 0

is the given MT.

Ellipsoid for delta chi^2=1: for 1D pdf and gaussian errors it is equivalent to +/- 1 sigma.

6 stations, source depth 10 kmPerturbing mechanism:

Strike = 106°, Dip = 11° , Rake = 153°Strike = 222°, Dip = 85° , Rake = 80°

VOL 9%

f < 0.2 Hz

Each point = a point inside the ellipsoid

of chi^2=1.

(colors unimportant)


(a6) … the largest parameter uncertainty

(a6)and how it is

made up

+++++

Singular vector almost parallel

to a6.

vectors are in coloumns, each row is one component

an extreme case of 1 station, source depth 60 km

1 station, source depth 60 km f < 0.2 Hz Perturbing mechanism: Strike = 106°, Dip = 11° , Rake = 153°Strike = 222°, Dip = 85° , Rake = 80°

VOL 9%

1 station, source depth 60 km

At least 2 vectors are ‘in the play’, with dominant components along a1, a4, a6



Perturbing mechanism: Strike = 106°

Dip = 11°Rake = 153°



Perturbing mechanism:Strike = 106°

Dip = 11°Rake = 153°

Attention !Even the MT itself is important…

We can apply the same ellipsoid to perturb a different MT[i.e. to study uncertainty of an earthquake of a different mechanism]

if the source position and stations remain the same.

(Reason: The problems is linear in a1, … a6.)


Perturbing anothermechanism199 74 -15

293 75 -163

Note a different uncertainty,

e.g. better rake …no problem like above)


Perturbing the former mechanism:Strike dip rake

106 11 153222 85 80

15380

marks a problem

Possible outlook

The cases may exist when MT can be determined from a very few stations

(Mars ??, forensic seismology??).

Possibility to design network extension, e.g. where to put efficient OBS.

Many factors are not included in this analysis, mainly the uncertainty of the crustal model.

And the noise! For example, when combining the land and ocean bottom stations,one should count with higher noise of the latter.

Etc. ….

Part 3Volume component

Great advantage of the MT inversion formulated by means of the basis mechanisms like in ISOLA: Note

that VOL is given by a single parameter, just a6. Mkk = a6.

6 basis mechanisms

Gallovič & Zahradník(submitted)

Multiple DC a single full MT(apparent VOL 16%

due to neglecting finite extent)

color green circles

We use synthetics of the single-source Myiagi as a model of the seismogram with a known VOL component

(although it is only apparent VOL).

= SIGMA from isola (incl. vardat) .257820E-01 .954336E-01 .415202E-01 .381398E-01 .438742E-01 .495832E-01

(a6) was already in ISOLA old, but here we learn about trade off between

different parameters within the 6D ellipsoid.

12 stations, f< 0.2 Hzdata variance=0.01 m2

VOL % : 16.4, DC % : 57.4, CLVD % : 26.2 strike,dip,rake: 206 32 101strike,dip,rake: 12 58 82

source depth 3.9 km

ISOLA11a…. vect.dat, sing.dat…. sigma.dat

(a6) … by far not the largest uncertainty

Trade off!

We know how to estimate the relative MT uncertainty (including VOL), because we know the 6D ellipsoid. What to do in case that we add two more free parameters (a7… depth, a8… source time) and the problem becomes non-linear?

Answer: The pdf has to be determined experimentally, and because VOL is given by just a single parameter, it is enough to construct a 1D pdf, just pdf (a6). This, however, will no more be possible without seismograms. Let’s use the synthetics with VOL=16%.

Even date-time: 20080510 15:56:20Displacement (m). Inversion band (Hz) 0.03 0.04 0.08 0.10

ObservedSynthetic

Gray w aveforms w eren't used in inversion.Blue numbers are variance reduction

-0.20

0.2NS

012

0 . 6 3 -0.20

0.2EW

0 . 7 7 -0.20

0.2Z

0 . 8 3

-0.10

0.101

10 . 9 3 -0.1

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-0.04-0.0200.020.04

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0 . 8 7 -0.04-0.0200.020.04

0 . 9 2 -0.04-0.0200.020.04

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0 . 2 0

coefficients of elem.seismograms a(1),a(2),...a(6): .589645E+19 .421103E+17 .120700E+20 .633459E+19 .199198E+20 -.484050E+19

moment (Nm): 2.790536E+19 moment magnitude: 6.9

VOL % : 16.4 DC % : 57.4

CLVD % : 26.2 strike,dip,rake: 206 32 101 strike,dip,rake: 12 58 82

= ZDÁNLIVÁ složka

pro bodový zdroj

How can the pdf be determined experimentally?

1) Inverting artificial seismograms produced for the expected parameter vector (=MT) by adding various realizations of „noise “.

[Can the noise substitute also the uncertainty of the crust?]

2) Exploring the parameter space in vicinity of the optimal solution (example: NonLinLoc).

If the 1D pdf is enough, such as pdf (a6), we can construct it by varying (perturbing) just a6, while

optimizing the remaining parametrs a1,…a5for each fixed a6. In an elegant way we can combine

linearity with respect to a1,…a5 (the least squares), and non-linearity in the depth a7 and time a8 (grid search).

Or, another alternative, proposed TODAY:

Our case of 1D pdf (a6): = 1 …. 1 degree of freedom

This is the theory behind the idea. (Numeric Recipes)

RECALL: Example of error volumes of a non-linear problem.Probabilistic location (NonLinLoc, A. Lomax)

V. Plicka

Epicenter (2 parameters)

Depth (1 single parameter may have a specific physical meaning).

Here 1D pdf (z) is of interest and can be found by searching at each depth the optimal horizontal position of the source.This is a good example of a 1D pdf !!!!

Highly important even in a relative sense

(without exact knowledge

of the data errors) !

Illustration how to get 1d pdf (still like if we know the ellipsoid, and have only a1,. …a6).Steps along a6 axes, searching optimum a1-a5 within the ellipsoid, recording its solution and min 2.

See next slide for the values.

0 -3 2 1 -2 -5 -.49583E+19 .58151E+00 214. 34. 117. 3. 60. 73. -1 -2 1 1 -2 -4 -.39667E+19 .72348E+00 211. 33. 112. 6. 60. 76. 0 -2 2 1 -1 -3 -.29750E+19 .80227E+00 212. 33. 112. 6. 60. 76. 0 -1 0 0 -1 -2 -.19833E+19 .88519E+00 210. 33. 107. 9. 58. 79. -1 0 -1 0 -1 -1 -.99166E+18 .93837E+00 205. 33. 101. 12. 58. 83. 0 0 0 0 0 0 .00000E+00 .10000E+01 206. 33. 102. 12. 58. 83. 1 0 1 0 1 1 .99166E+18 .93837E+00 207. 32. 103. 12. 58. 82. 0 1 0 0 1 2 .19833E+19 .88519E+00 203. 32. 96. 15. 58. 86. 0 2 -2 -1 1 3 .29750E+19 .80227E+00 198. 33. 89. 19. 57. 90. 1 2 -1 -1 2 4 .39667E+19 .72348E+00 199. 32. 90. 19. 58. 90. 0 3 -2 -1 2 5 .49583E+19 .58151E+00 195. 33. 84. 22. 58. 94.

delta (a6) delta 2 strike dip rake

Note how, increasing delta (a6), the misfit delta increases.

See also variations of the strike, dip, rake.

Position inside ellipsoid

pdf= exp(-0.5 ) )/( sqrt(2))

Pdf_theor= exp(-0.5 x2)/( sqrt(2))x=(a6)/

Transition from misfit to pdf (green diamonds)and comparison with theoretic 1D pdf (curve)

What remains to be done is to construct the 1D pdf without the

ellipsoid, i.e. using data (seismograms).

Fix a given a6 and search the optimum a1,…a5.

ISOLA new:

A) Perturbing the MT solution within the error ellipsoid.

B) Experimental determination of the 1D pdf (a6) for VOL.

INV1_subtractVOL_thenFULL.dat Original data: coefficients of elem.seismograms a(1),a(2),...a(6): .589645E+19 .421103E+17 .120700E+20 .633459E+19 .199198E+20 -.484050E+19 Inversion result after subtracting VOL and making again FULL MT inversion: coefficients of elem.seismograms a(1),a(2),...a(6): .590122E+19 .767714E+17 .120507E+20 .631315E+19 .199242E+20 .284984E+17 moment (Nm): 2.725186E+19 moment magnitude: 6.9 VOL % : .1 DC % : 68.7 CLVD % : 31.2 strike,dip,rake: 206 32 101

INV1_subtractVOL_thenDEVIA.dat

Original data: coefficients of elem.seismograms a(1),a(2),...a(6): .589645E+19 .421103E+17 .120700E+20 .633459E+19 .199198E+20 -.484050E+19

Inversion result after subtracting VOL and making DEVIA MT inversion: coefficients of elem.seismograms a(1),a(2),...a(6):

.589858E+19 .458258E+17 .120573E+20 .631724E+19 .199108E+20 .000000E+00


ObservedSynthetic

Gray waveforms w eren't used in inversion.Blue numbers are variance reduction

-0.100.1

NS01

2- 2 . 2 6 -0.10

0.1EW

- 7 . 6 6 -0.100.1

Z

- 1 . 3 8

-0.10

0.1

011

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00.1

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0 . 2 0 -0.04-0.0200.020.04

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a6=-0.4e19 Interesting byproduct : We see seismograms of the volume component.! !

VOL=red

complete=black


ObservedSynthetic

Gray waveforms w eren't used in inversion.Blue numbers are variance reduction

-0.100.1

NS01

20 . 9 4 -0.10

0.1EW

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a6=+0.4e19

The resulting pdf (a6), without knowing the singular vectors, found experimentally from seismograms,including LSQ for a1,….a5 and the grid search for source depth and time.

Very preliminary.Not checked enough.

Concluding remarks I• Analysis of singular vectors makes it possible to understand

when and why sometimes just few stations are sufficient for MT. • Application in designing new station networks.• Application in forensic seismology. • Outlook = Portugal (Susana Custódio) = Greece, Trichonis Lake (Dana Křížová)

Dana already documented cases where 1 station is enough, giving almost same strike, dip, rake as 10 stations for a shallow event.)

Concluding remarks II• Advantage of the MT representation by a1-a6 (the volume component being described by single

parameter, a6).• It makes it sense to study a one-dimensional case, pdf (a6).• Pdf (a6) for a linear problem is known (ellipsoids).• Pdf (a6) for a non-linear problem is easy to find experimentally (linear part in MT by LSQ, and the

non-linear part in position and time by grid search.• Outlook –Santorini Island sequence with a preliminary

detection of a reliable VOL component (Dana Křížová).

A bit more ?

Another use of singular vectors• To interpret a given focal mechanism (a1,…..a6) from

‘viewpoint’ of a given station network (to express the parameter vector by means of its

decomposition into singular vectors for a given network). aest=VVT a=R a

• To understand the role of small singular values, for example, by means of regularization.

Caution: We do not need the regularization. But we can try it as a tool to artificially bias the solution along the most vulnerable directions. In this sense the regularization serves instead of high noise; data are not used!)

ISOLA new:

A) Perturbing the MT solution within the error ellipsoid.

B) Experimental determination of the 1D pdf (a6) for VOL.

C) Analysing decomposition of MT into singular vectors for better understanding of the

dangerous effects of small singular values.

Portugal: 1 station, source depth 10 km.

VOL 8.6% 106. 11. 153.

1 2 3 4 5 6com ponent of the param eter vector

-4E +015

-2E +015

0

2E+015

4E+015

co lor = ind ividua l s ingu lar vectors1 b lack, 2 b lue, 3 green4 ye llow , 5 red, 6 p ink

singular values: .323132E-10 .221424E-10 .210062E-10 .776380E-11 .503142E-11 .128343E-11

Bold crosses: a given vector (a1, …a6). Each of its component depend in general

on all singular vectors.

Small data errors would be most strongly amplified along V6.

Let’s illustrate it in a drastic way – instead

of amplifying error,simply cancel V6.

“The bold crosses are summed up from color crosses”


Regularization:

Excluding w<wmax/ 10zeroing V6

VOL 9% 106. 11. 153.VOL 13% 120. 7. 167.


The regularized solution is shown by squares. Note difference with respect to bold crosses.

Effect is small since ‘pink’ components were small.


-4E+015

-2E+015

0

2E+015

4E+015

color = ind ividua l s ingular vectors1 b lack, 2 b lue, 3 green4 ye llow , 5 red, 6 p ink


VOL 9% 106. 11. 153.VOL 13% 120. 7. 167.


Satbility because vector of parameters was close to one (yellow)

singular vector!


-4E+015

-2E+015

0

2E+015

4E+015

color = ind ividua l s ingular vectors1 b lack, 2 b lue, 3 green4 ye llow , 5 red, 6 p ink


Same regularizationhere having greater effect:

Excluding w<wmax/ 10zeroing V5, V6

VOL 9% 106. 11. 153. VOL 3% 146. 6. -168.


Decomposition of the parameter vector is more variable than before.

This is bad, since with noisy data the imperfection of the components

will more easily harmthe solution.


-4E+015

-2E+015

0

2E+015

4E+015

color = ind ividua l s ingu lar vectors1 b lack, 2 b lue, 3 green4 ye llow , 5 red, 6 p ink


Excluding w<wmax/ 10zeroing V5, V6

VOL 9% 106. 11. 153. VOL 3% 146. 6. -168.


Regularized = squares. Note difference with respect to bold crosses.


-4E+015

-2E+015

0

2E+015

4E+015

color = ind ividua l s ingu lar vectors1 b lack, 2 b lue, 3 green4 ye llow , 5 red, 6 p ink

Portugal: 6 stations, source depth 10 km.


-4E+015

-2E+015

0

2E+015

4E+015

co lor = ind ividua l s ingu lar vectors1 b lack, 2 b lue, 3 green4 yellow , 5 red, 6 p ink Problem better posed.

To see any effect, we FORMALLY need a stronger regularization.

Excluding w<wmax/ 3zeroing V6

VOL 9% 106. 11. 153. VOL 5% 104. 11. 151. .


A highly stable case.



-4E+015

-2E+015

0

2E+015

4E+015

co lor = ind ividua l s ingu lar vectors1 b lack, 2 b lue, 3 green4 yellow , 5 red, 6 p ink

Excluding w<wmax/ 3 zeroing V4, V5, V6

VOL 9% 106. 11. 153. VOL 1% 53. 28. 152.


A highly unstable case.


max/min = 3zeroing V4, V5, V6

VOL 9% 106. 11. 153. VOL 1% 53. 28. 152.



-4E +015

-2E +015

0

2E+015

4E+015

co lor = ind iv idual s ingu lar vectors1 b lack, 2 b lue, 3 green4 ye llow , 5 red, 6 p ink


-4E +015

-2E +015

0

2E +015

4E +015

co lor = ind ividual s ingu lar vectors1 b lack, 2 b lue, 3 green4 ye llow , 5 red, 6 p ink

VOL=49% 106. 11. 153.

1 2 3 4 5 6com ponent o f the param eter vector

-4E +015

-2E +015

0

2E+015

4E+015

color = ind iv idual s ingu lar vectors1 b lack, 2 b lue , 3 green4 ye llow , 5 red , 6 p ink

Portugal: 6 stations, source depth 60 km,but a large VOL component.

VOL=7 % 102. 14. 148.


-4E +015

-2E +015

0

2E+015

4E+015

co lor = ind iv idua l s ingu lar vectors1 b lack, 2 b lue, 3 green4 ye llow , 5 red, 6 p ink

Zeroing V6 with the least singular value strongly changed only a6.Reason: The zeroed singular vector was almost parallel with a6.

Caution, do not misinterpret with DEVIA solution.But the message is analogous:

Large VOL added to a DC does not implythat without removing VOL we obtain a wrong DC solution.

singular values:

.668926E-11 .472489E-11 .366529E-11 .212083E-11 .186006E-11 .873139E-12

singular vectors V -.392628E+00 .148396E+00 .390792E+00 .512404E+00 -.631546E+00 -.984670E-01 -.154920E+00 -.557097E+00 -.324050E+00 .680151E+00 .310686E+00 .387815E-01 .240984E+00 .778522E+00 -.261410E+00 .457367E+00 .241377E+00 .682164E-02 .577032E+00 -.173998E+00 -.474157E+00 .572759E-01 -.634798E+00 -.753926E-01 .656356E+00 -.175597E+00 .670112E+00 .248769E+00 .165035E+00 .137694E-01 .171067E-03 .203813E-01 .792361E-02 .220399E-01 -.127106E+00 .991403E+00

6th vector V6 almost parallel to a6

estimating uncertainty of moment tensor using singular vectors

Documents

source depth

data errors

source position

gaussian case

mt solution

given mt

mt components

true mt