Estimating uncertainty of moment tensor using singular vectors. J. Zahradník Charles University in Prague Czech 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 uncertainty by singular vectors in 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! X X

Part 2 Linear problem for MT (position and source time fixed) Motivation: Greece -Dana Křížová Portugal - Susana Custódio

Uncertainty in practice: 10/5/ 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 star delta a i = 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 km Perturbing 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)

6 stations, source depth 10 km  (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

1 station, source depth 10 km 1 station, source depth 60 km Perturbing mechanism: Strike = 106° Dip = 11° Rake = 153°

6 stations, source depth 10 km 6 stations, source depth 60 km 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.)

1 station, source depth 10 km Perturbing another mechanism Note a different uncertainty, e.g. better rake … no problem like above) 1 station, source depth 10 km Perturbing the former mechanism: Strike dip rake 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 3 Volume 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. M kk = 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) E E E E E E-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 Hz data variance=0.01 m 2 VOL % : 16.4, DC % : 57.4, CLVD % : 26.2 strike,dip,rake: strike,dip,rake: 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%.

coefficients of elem.seismograms a(1),a(2),...a(6): E E E E E E+19 moment (Nm): E+19 moment magnitude: 6.9 VOL % : 16.4 DC % : 57.4 CLVD % : 26.2 strike,dip,rake: strike,dip,rake: = 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,…a5 for 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.

E E E E E E E E E E E E E E E E E E E E E E 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 x 2 )/(  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): E E E E E E+19 Inversion result after subtracting VOL and making again FULL MT inversion: coefficients of elem.seismograms a(1),a(2),...a(6): E E E E E E+17 moment (Nm): E+19 moment magnitude: 6.9 VOL % :.1 DC % : 68.7 CLVD % : 31.2 strike,dip,rake:

INV1_subtractVOL_thenDEVIA.dat Original data: coefficients of elem.seismograms a(1),a(2),...a(6): E E E E E E+19 Inversion result after subtracting VOL and making DEVIA MT inversion: coefficients of elem.seismograms a(1),a(2),...a(6): E E E E E E+00

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

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). a est =VV T 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% singular values: E E E E E E-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”

Portugal: 1 station, source depth 10 km. Regularization: Excluding w<wmax/ 10 zeroing V6 VOL 9% VOL 13% singular values: E E E E E E-11 The regularized solution is shown by squares. Note difference with respect to bold crosses. Effect is small since ‘pink’ components were small.

Portugal: 1 station, source depth 10 km. VOL 9% VOL 13% singular values: E E E E E E-11 Satbility because vector of parameters was close to one (yellow) singular vector!

singular values: E E E E E E-11 singular vectors V E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 expanding m into eigenvectors coeficients: E E E E E E+15 coeficients * sing.values E E E E E E+04 coef(1)* V(1), coef(2)*V(2).....coef(6)*V(6) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+15 true parameters E E E E E E+15 vol1, vol2, absvol2, strike, dip, rake.13278E E E a1= a2= a1= a2= a est = V V T a = a

====== REGULARIZATION ============= ratio max/min sing values= 10 eigenvector set to zero for j= 6 regularized eigenvectors: E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 expanding mest into eigenvectors coeficients E E E E E E+00 coeficients * sing.values E E E E E E+00 coef(1)* V(1), coef(2)*V(2).....coef(6)*V(6) E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 estimated parameters E E E E E E+15 vol1, vol2, absvol2, strike, dip, rake.35767E E E a est = Vp Vp T a = R a

Portugal: 1 station, source depth 60 km. Same regularization here having greater effect: Excluding w<wmax/ 10 zeroing V5, V6 VOL 9% VOL 3% singular values: E E E E E E-12 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 harm the solution.

Portugal: 1 station, source depth 60 km. Excluding w<wmax/ 10 zeroing V5, V6 VOL 9% VOL 3% singular values: E E E E E E-12 Regularized = squares. Note difference with respect to bold crosses.

Portugal: 1 station, source depth 60 km. VOL 9% VOL 3% singular values: E E E E E E-12

Portugal: 6 stations, source depth 10 km. Problem better posed. To see any effect, we FORMALLY need a stronger regularization. Excluding w<wmax/ 3 zeroing V6 VOL 9% VOL 5% singular values: E E E E E E-11 A highly stable case.

Portugal: 6 stations, source depth 60 km. Excluding w<wmax/ 3 zeroing V4, V5, V6 VOL 9% VOL 1% singular values: E E E E E E-12 A highly unstable case.

Portugal: 6 stations, source depth 60 km. max/min = 3 zeroing V4, V5, V6 VOL 9% VOL 1% singular values: E E E E E E-12

VOL=49% Portugal: 6 stations, source depth 60 km, but a large VOL component.

VOL=7 % 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 imply that without removing VOL we obtain a wrong DC solution.

singular values: E E E E E E-12 singular vectors V E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+00 6 th vector V6 almost parallel to a6