appendix to the paper ”toward understanding slip-inversion...

Appendix to the paper

”Toward understanding slip-inversion uncertainty andartifacts II: singular value analysis”by F. Gallovic and J. Zahradnık

A1 Smoothing constraint

The smoothing constraint represents an alternative regularization approach to the one pre-sented in the main text, i.e. truncation by means of singular value decomposition. Thespace-time smoothing is introduced by adding new lines to matrix G. The lines representlinear equations mi − mj = 0, where indices i and j correspond to all combinations of theadjacent model parameters in the x-t plot. All these additional lines are multiplied by factorLw = wML−1

s /2 = wµST/2M0, where S and T are the fault area and the time interval,respectively. Parameter w > 0 is subject to change, controlling the relative weight of thesmoothing constraint; all other parameters are defined in the main text. Here we show theeffects of smoothing for two methods of the least-squares inversion: i) the singular value de-composition (SVD) as in the main text (since smoothing is applied, all singular vectors inthe solution expansion can be used) and ii) the non-negative least-squares (NNLS) methodproviding results with the positivity constraint on the slip velocity field.

A1.1 Synthetic model

Figure A1A shows the variance reduction as a function of smoothing weight w for the syntheticbilateral model using the SVD. Note the prohibitively large drop of the variance reduction forw > 0.1. Figure A2 displays the inversion results for selected smoothing weights (see legend).As compared with the main text without smoothing, one can see that the spurious asperityis even emphasized when the smoothing is applied; the spurious asperity exists even for weaksmoothing, w = 0.001, with a very large variance reduction of 0.999 (the left panel in FigureA2). Figure A3 shows the same smoothing, but the inversion is performed using the NNLSmethod. Note that the main characteristics of the inverted model are the same as withoutthe positivity constraint, including the false asperity in the fault center.

0.25

0.5

1

0.001 0.01 0.1 1

Var

ianc

e re

duct

ion

Smoothing weight

A)

0.001 0.01 0.1 1

Smoothing weight

B)

Figure A1. Dependence of the variance reduction on the smoothing weight for the synthetic bilateral

source model (A) and for the real Movri Mountain earthquake data (B). The SVD approach is used,

employing all singular vectors (i.e. no further regularization is considered).

1

w=0.001 VR=0.999

0 10 20 30 40Along strike (km)

0

5

10

15T

ime

(s)

w=0.010 VR=0.995


w=0.100 VR=0.834


-0.02

0

0.02

0.04

0.06

Slip

vel

ocity

(m

/s)

Figure A2. Inverted slip velocity distribution for the synthetic bilateral source model solved by the

SVD method when the smoothing constraints are applied. VR and w represent the variance reduction

and the smoothing weight, respectively. Note that the false asperity in the center of the fault persists

and cannot be avoided by smoothing (compare with the input model in Figure 2A in the main text).

w=0.001 VR=0.999


0

5

10

15

Tim

e (s

)

w=0.010 VR=0.991


w=0.100 VR=0.823


-0.02

0

0.02

0.04

0.06

Slip

vel

ocity

(m

/s)

Figure A3. Same as Figure A2, but the inversion is performed using the non-negative least-squares

method (NNLS). Similarly to Figure A2, the false asperity in the center of the fault is still present.

A1.2 Real data

Figure A1B shows the decrease of the variance reduction with increasing smoothing-constraintweight. Similarly to the synthetic model, we should keep w below 0.1, after which the vari-ance reduction is too low. Figure A4 displays the resulting slip velocity x-t plots for selectedsmoothing weights solved by the SVD approach. Similarly, Figure A5 shows the slip velocitymodels obtained by the NNLS method for the same weights.

2

w=0.010 VR=0.727


0

5

10

15T

ime

(s)

w=0.050 VR=0.680


w=0.100 VR=0.611


-0.02

0

0.02

0.04

Slip

vel

ocity

(m

/s)

Figure A4. Same as Figure A2 (SVD approach), but for the real Movri Mountain data. Note that

the minimum weight w = 0.01 (left) is larger than in Figure A2, because the smaller weights already

yield an excessively distorted inverted model for real data.

w=0.010 VR=0.707


0

5

10

15

Tim

e (s

)

w=0.050 VR=0.675


w=0.100 VR=0.609


-0.02

0

0.02

0.04

Slip

vel

ocity

(m

/s)

Figure A5. Same as Figure A4, but the inversion is performed using the NNLS approach. Note that

the result for w = 0.01 (left) is apparently more realistic than in Figure A4.

A2 Data weighting

Here we show the influence of the data weighting on the slip inversion. The weights are appliedto each station component individually, being reciprocals of the L2 norm of the respectiveseismogram (displacement time series in the studied frequency range). This modifies matrixG and hence also the corresponding singular vectors (see Figure A6 for the whole networkconsidered). One can see that the leading singular vectors have a different structure thanthose presented in the main text without weighting (Figure 4). In particular, already thefirst singular vectors are shaped in patches rather than in uniform strips, as expected due todown-weighting of the strongly dominant directive station (SER).

Figure A7 shows the c = GTd vector, assuming all stations and including the station-

component weights, for both the synthetic bilateral model (A) and the real case of the MovriMountain earthquake (B). Note that the shape of GT

d is very similar to the shape of thesecond leading singular vector (Figure A6). The effects of the weighting upon the inversionresults are then demonstrated in Figures A8 (synthetic bilateral model) and A9 (Movri Moun-tain data). Note the smaller variance reductions in the case of the Movri Mountain data ascompared to those presented in the main text (Figure 8). This is attributed to down-weightingthe closest stations in the inversion.

3

λ(1)=12.720 λ(2)=11.788 λ(3)=11.343 λ(4)=10.704 λ(5)=9.431

λ(6)=9.328 λ(7)=8.320 λ(8)=7.633 λ(9)=7.203 λ(10)=6.244

λ(20)=2.326 λ(30)=0.700 λ(40)=0.233 λ(50)=0.147 λ(60)=0.102


0

5

10

15

Tim

e (s

) λ(70)=0.076 λ(80)=0.060

-0.1 0 0.1Amplitude (dimensionless)

λ(90)=0.046 λ(100)=0.038 λ(110)=0.029

Figure A6. Singular vectors for the whole station network (Figure 1); analogy to Figure 4, but

applying weights to the individual station components. Starting already at λ(2), the singular-vector

pattern includes both ‘inclinations’ typical for the individual stations (see Figure 2 in the main text).

This property explains why the slip inversion results (Figures A8-A9) differ from those in the main

text.

A)

0 10 20 30 40

Along strike (km)

0

5

10

15

Tim

e (s

)

-2

-1

0

1

2

3

4GTd (s/m)

B)

0 10 20 30 40

Along strike (km)

-2

-1

0

1

2

3GTd (s/m)

Figure A7. A) The GTd vector for the synthetic test similar to Figure 5 in the main text, including

the whole network, but assuming station-component weights. Compared to Figure 5, the SER station

is down-weighted relatively to the rest of the stations, which results in a more symmetric pattern. B)

Same as A, but for the real Movri Mountain data (to be compared with Figure 7 in the main text).

4

SER

λmin=

λmax/5

ZAK RGA All 8 stations

λmin=

λmax/10

λmin=

λmax/100

λmin=

λmax/1000


0

5

10

15

Tim

e (s

)

-0.05 0 0.05 0.1Slip velocity (m/s)

Figure A8. Truncated solutions of the slip inversion problem for the synthetic bilateral model (Figure

2A in the main text); station-component weighting is employed. The figure is to be compared with

Figure 6 in the main text.

5

SER

λmin=

λmax/3

ZAK RGA All 8 stations

0.464

λmin=

λmax/50.473

λmin=

λmax/70.477

λmin=

λmax/90.479

λmin=

1/20λmax


0

5

10

15

Tim

e (s

)

0 0.05Slip velocity (m/s)

0.500

Figure A9. Same as Figure A8, but for the real Movri Mountain data, to be compared with Figure

8 of the main text. The numbers in the rightmost panels are the corresponding variance reductions.

Note that they are lower than in the case when no weights are applied (mostly due to the worse

fit of the SER station in the present case). This down-weighting of station SER resulted in a more

pronounced but still very uncertain indication of the rupture propagation from 10 km to 0 km at 5 to

10 seconds.

6

appendix to the paper ”toward understanding slip-inversion...

Documents