Inversion of Earthquake Rupture Process:Theory and Applications

Transcription

1 Iversio of Earthquake Rupture Process:Theory ad Applicatios Yu-tai CHEN 12 * Yog ZHANG 12 Li-sheg XU 2 1School of the Earth ad Space Scieces, Pekig Uiversity, Beijig Istitute of Geophysics, Chia Earthquake Admiistratio, Beijig 181 *Correspodet author; cheyt@cea-igp.ac.c

2 Iversio of Earthquake Rupture Process:Theory ad Applicatios 2. Theory ad Techiques

3 2.1 Seismic Iversio

4 Groud surface Dip Iversio

5 2.1.1 Iversio with Rake Fixed P The seismic waves radiated from a fiite-fault ca be represeted as U ( P, t) G ( P, t; Q,) m ( Q, t) d' p, q pq If a purely shear dislocatio is assumed O R r Q dσ Σ R m ( Q, t) M ( Q, t)( e v e v ) pq p q q p The we have U ( P, t) G ( P, t; Q,) [ M ( Q, t)( e v e v )] d' p, q p q q p

6 If assume U ( P, t) G ( P, t; Q,) [ M ( Q, t)( e v e v )] d' p, q p q q p g ( P, t; Q,) G ( P, t; Q,) ( e v e v ) p, q p q q p It becomes U ( P, t) g ( P, t; Q,) M ( Q, t) d' The seismograms ca be recorded durig a earthquake, the Gree s fuctios ca be calculated with a preferred earth model, the ukows are the seismic momet fuctios, which are depedet o place ad time.

7 U ( P, t) g ( P, t; Q,) M ( Q, t) d' e R P For a case of spatial poit source (R>>L), L is the source dimesio), a path approximatio ca be assumed R R g ( P, t; Q,) g ( P, t ; O,) Where Whe = R' R c ' R 2 R' R L, = = - 2 c r e c R O r Q dσ Σ L

8 With the path approximatio It ca be rewritte as U ( P, t) [ g ( P, t ; O,) M ( Q, t)] d' U ( P, t) g ( P, t; O,) [ ( t ) M ( Q, t)] d' The itegratio o the right is the apparet source time fuctio (ASTF) S P t t M Q t d (, ) A [ ( ) (, )] ' So two coditios are demaded for the ASTF R L At far-field distaces = R' R c For sigle phase

9 To perform the iversio, it is ecessary to discretize the fault plae ito sub-faults 1 2 k-1 3 k-k r k k+k r k+1 k r L S Sice the calculatios of Gree s fuctios is doe based o poit sources, a poit source approximatio is eeded for the sub-faults. It meas the sub-fault size ca be eglectable. It demads R >> L S >>L S Spatial poit source Temporal poit source

10 With the poit source approximatio, the itegratios o the fault ca be replaced as summatios U ( P, t) [ g ( P, t ; O,) M ( Q, t)] d' U ( P, t) g ( P, t; O,) [ ( t ) M ( Q, t)] d' S P t t M Q t d (, ) A [ ( ) (, )] ' They are U ( t) [ g ( t ) M ( t)] m m mk k k m m m U ( t) g ( t) S ( t) A S ( t) [ ( t ) M ( t)] m mk k A k

11 Two iversio ways ca be classified I. ASTF iversio: get the ASTFs by decovolvig the Gree s fuctio from the seismograms, ad the ivert the ASTFs for rupture model m m m U ( t) g ( t) S ( t) A S ( t) [ ( t ) M ( t)] m mk k A k II. Seismogram iversio U ( t) [ g ( t ) M ( t)] m m mk k k

12 1 Apparet Source Time Fuctio Iversio S ( t) [ ( t ) M ( t)] m mk k A k The matrix equatio [S δ A] [ ][M ] Where [δ] cosists of block matrixes δ δ δ [ ] δ δ =... δ K B B B 21 B MK δb M 1 MK B δb Ad the block matrixes [δ B ] is 1 1 [ δb ( t)] [ δ B ( t 1)] [ δb ( t 1)]... 1

13 . A example of the matrix. Blue dots deote the o-zero elemets

14 2 Waveform Iversio U ( P, t) [ g ( P, t ; O,) M ( Q, t)] d' The matrix equatio [U] [g][m ] Where [g] cosists of block matrixes K g g... g K g g... g [g] mk g... M 1 M 2 MK g g... g Whe the legth of sub-fault STF is shorter tha that of the Gree s fuctio Ad the block matrixes [g B ] is mk g () 1 mk mk g ( 2) g ( 1) mk [g ( t)] mk mk mk g ( Lg )... g ( 2) g ( 1)

15 1.2 Iversio with Rake Variatio U ( P, t) G ( P, t; Q,) m ( Q, t) d' p, q pq The slip vector o the fault e ca chage with time m ( Q, t) M ( Q, t)[ e ( t) v e ( t) v ] pq p q q p The U ( P, t) G ( P, t; Q,) { M ( Q, t) [ e ( t) v e ( t) v ]} d' p, q p q q p

16 U ( P, t) G ( P, t; Q,) { M ( Q, t) [ e ( t) v e ( t) v ]} d' p, q p q q p It ca be writte as U ( P, t) [ g ( P, t; Q,) M ( Q, t) g ( P, t; Q,) M ( Q, t)] d' Where g ( P, t; Q,) G ( P, t; Q,) v, M ( Q, t) M ( Q, t) e ( t) 1 1, g ( P, t; Q,) G ( P, t; Q,) v, M ( Q, t) M ( Q, t) e ( t) 2 2, The iversio equatio is [U] M 1 [g1 g 2] M 2

17 Strike 2 rake =9 3 ormal directio 1 rake =

18 The iversio equatios ASTF iversio Seismogram iversio with rake fixed [U] [g][m ] [S δ A] [ ][M ] Seismogram iversio with rake variatio [U] M 1 [g1 g 2] M 2 The ukow parameters i the 3 equatios are the history of scalar seismic momet. Whe the Gree's fuctios are calculated by cosiderig the step respose, the ukows become the momet rate fuctio (source time fuctio). The ukows of momet rate ca be trasferred ito fault slip-rate by multiplyig the Gree's fuctios by the shear modulus ad the sub-fault area.

19 The iversio equatios ASTF iversio Seismogram iversio with rake fixed [S A] [ δ ][s] [U] [g][s] Seismogram iversio with rake variatio s 1 [U] [g1 g 2] s 2

20 2.1.3 Some limitatios or costraits are eeded to stabilize the solutio Limitatio of the maximum rupture velocity Limitatio of the maximum rupture duratio of the sub-fault Limitatio of o-egative solutio 1 Fault plae Sub-fault STF D 2 k 3 r r D A sketch of the rupture iitiatio time (τ r ) ad rupture duratio (D) of a subfault STF

21 1 Spatial smoothig k k1 k1 kkr kkr 4 s ( t) [ s ( t) s ( t) s ( t) s ( t)] [D][s] [ ] 2 k-1 3 k-k r k k+k r k+1 k r Temporal smoothig 2 s k ( t) [ s k ( t 1) s k ( t 1)] [T][s] [ ] Scalar momet miimizatio k s () t [Z][s] [ ]

22 2.1.4 Equatios for the three kids of iversios The iversio equatios ASTF iversio Seismogram iversio with rake fixed Seismogram iversio with rake variatio S A δ D [s] 1 2 T U g 1 D [s] 2 T Z 3 g1 g2 D U 1 D s 1 T 2 s 2 T Z 3 Z

23 2.1.5 A Example: The 29 M W 6.3 L'Aquila earthquake

24 A example: The 29 M W 6.3 L'Aquila earthquake Teleseismic statios Azimuth-depedet ASTFs

25 Fault slip distributios STFs ASTF iversio Seismogram iversio with rake fixed Seismogram iversio with rake variatio

26 ASTF iversio Seismogram iversio with rake fixed Seismogram iversio with rake variatio

27 Retrieved ad sythetic ASTFs ASTF iversio Seismogram iversio with rake fixed Seismogram iversio with rake variatio

28 2.2 Joit Iversio of seismic ad geodetic data Seismic data (Wag et al., 211) GNSS data ISAR Data

29 Fault slip iversio with geodetic data whe cosiderig the rake variatio [E] f 1 [B1 B 2] f 2 The fault slip of each sub-fault is the summatio of slip-rate f1 J s1 f J s 2 2 The matrix [J] is a sparse matrix, i which the o-zero elemets equal [J] Sub-fault 1 Sub-fault 2 Sub-fault K

30 With the equatios [E] f 1 [B1 B 2] f 2 f1 J s1 f J s 2 2 The relatio betwee the deformatio data ad the fault slip-rate is [E] s 1 [H1 H 2] s 2 Where [H 1] [B 1] [J], [H 2] [B 2] [J]

31 g g D U D T s T s Z E Z H H s [E] [H H ] s g g D U D s T s T Z Z The equatios of seismic ad geodetic data iversio for slip-rate are The joit iversio ca be performed by solvig

32 . The differece i values of the seismic data U ad the deformatio data E may reach several orders of magitude, a ormalizatio is ecessary to esure that they are comparable i the joit iversio. Usually we ormalize them by their square root of eergy, It makes the two datasets be equally weighted i the least-square optimizatios if 1 U E U E 2 E dt 2 U dt

33 A example: The 29 M W 6.3 L'Aquila earthquake Seismic iversio ISAR iversio Joit iversio

34 Seismic iversio ISAR iversio Joit iversio Fault slip distributios of the 29 L'Aquila earthquake obtaied by teleseismic data iversio (a), ISAR data iversio (b), ad joit iversio (c). Give model Seismic model ISAR model Joit model

35 A geeral tests of idividual data iversio ad joit iversio. Seismic data teds to recover the slip patch which released more momet, while geodetic ISAR data maily costrai the shallow slips. The joit iversio sythesize the advatages of the two datasets. Give model Seismic model ISAR model Joit model

36 Summary We have described the theory ad techiques of iversio of earthquake rupture process, ad discussed the limitatios ad costraits existed i the seismic iversio. Take the 29 M W 6.3 L'Aquila earthquake as a example to ephamcize the eeds i joit iversio usig both of the seismic ad geodetic.

37 谢谢! Thak you! Спасибо!