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, Greece Cooperation: D. Křížová, V. Plicka, E. Sokos, A. Serpetsidaki, G-A. Tselentis

ITSAK, Greece 2 victims hundreds of injuries

HYPO and DD relocation: A. Serpetsidaki, Patras PSLNET BB and SM (SER, MAM, LTK, PYL co-operated by Charles Univ.) ITSAK SMNOA 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)

Model setup 2 asperities along the fault ( azimuth 30°) rupture velocity: Vr = 3 km/s 3 scenarios of rupture propagation: from the left, right and middle 8 stations: as for the earthquake crustal model: 1-D; Haslinger et al., 1999 full-wave synthetics, 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. x moment

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

Iterative method (F.G.) x t

position time cumul. mom. varred E E E E E E E E E E+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-source at that respective x-t node point, with moment adjustment. Varred = c 2, where c is the correlation. For simplicity, for c 2, 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-source at that respective x-t node point, with moment adjustment. Varred = c 2, where c is the correlation For simplicity, for c 2, we often use term ‘correlation’. t waveform at a station fitting with Mo=1 adjusting Mo

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.

Idea 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. Most likely x-t points (larger correlation) are those common to all stations. [analogy with back-projection methods]

Idea 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, it is mapped onto x-t plane (at all possible x-t positions)... fitting with Mo=1 adjusting Mo Varred is evaluated from the whole seismogram. Varred informs about ‘signal detection’. (multiplication by a constant)

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 stations 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. x t Signals from here are not supported by ZAK and SER.

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

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) The ‘bright spots’ relate with asperities, but a very cautious interpretation is needed.

3 stations

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.

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 ?

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. Strip of ZAK ? Strip of SE5 ?

Indeed, seismograms enable reconstruction of the single-station strips: SE5 ZAK 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) fitting with Mo=1 adjusting Mo Varred is HIGH because the strong pulse is fitted Varred is LOW because the weak pulse is fitted more distant asperity provides a weaker signal

Single station: ZAK E E+00 fitting closer asperity

Single station: ZAK E E+00 fitting distant asperity

Single station: ZAK E E E E+00

Rupture propagation from the right Sum of 3 SE5 ZAK RGA

Rupture propagation from the right Sum of 3 SE5 ZAK RGA corr01

Rupture propagation from the right Sum of 3 corr01

Rupture propagation from the right Sum of 3 corr01 = 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 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 = false

Partial results We 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 - continued Directive/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 - outlook How 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’ ISOLA whose 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 f < 0.2 Hz

MAM ZAK f < 1 Hz

all stations, f < 0.2 Hz all stations; f < 1.0 Hz Andravida correlation plots

Any other implication for Andravida ? B) We try ISOLA with different finite sub1, for example: without a finite sub1 with a finite sub1

Conclusions In 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 other horizontally and vertically f < 1 Hz f < 0.2 Hz Station distribution: as for the Andravida earthquake