FRÉCHET KERNELS AND THE ADJOINT METHOD

Transcription

1 PART II FRÉCHET KERNES AND THE ADJOINT METHOD 1. Setp of the tomographic problem: Why gradients? 2. The adjoint method 3. Practical 4. Special topics (sorce imaging and time reversal)

2 Setp of the tomographic problem: Why gradients?

3 Problem setp Find an Earth model m sch that a sitably defined misfit X is minimal. The nmber of model parameters and the nmerical cost of the forward problem prevent the application of probabilistic methods. The minimisation proceeds iteratively: 1. Start from initial Earth model 2. Update according to mi 1 mi i hi with ( mi 1) ( mi ) step length descent direction 1.1

4 Problem setp 1. Start from initial Earth model 2. Update according to mi 1 mi i hi with ( mi 1) ( mi ) step length descent direction 1.1 h i m 1.2 The family of gradient methods: method of steepst descent: h i conjgate-gradient methods Newton and Newton-like methods variable-metric methods / m

5 Problem setp 1. Start from initial Earth model 2. Update according to mi 1 mi i hi with ( mi 1) ( mi ) step length descent direction 1.1 h m m 1

6 Problem setp 1. Start from initial Earth model 2. Update according to mi 1 mi i hi with ( mi 1) ( mi ) step length descent direction 1.1 Iteratively approach the minimm misfit by following the local descent directions. m m 1 m 2 m 3 m 4

7 Problem setp The fll gradient with all its components - is needed in each iteration. The most straightforward approach: approimate the gradient by finite-differences: ( m) (..., mk h,...) (..., m m h k k,...) 1.3 Eample with 5, model parameters: 5,1 forward simlations.5 h per simlation 126 processors 5 earthqakes 4 simlations per conjgate gradient iteration 1 conjgate gradient iterations 6.3e 1 cp hors 72, cp years

8 Problem setp SOUTIONS: Atomatic differentiation (AD, - Differentiation of compter programmes - Atomatic bt inefficient becase ignorant abot physics Adjoint method - Developed in optimal control theory (J.-- ions, 196s) - Epress gradients in terms of forward and adjoint fields - Can be a bit tedios, bt is fll of interesting physics.

9 1. Prelde: Misfits and Fréchet derivatives 2. The adjoint trick 3. Eamples 4. The adjoint wave eqation

10 Prelde I: Fréchet and classical derivatives The adjoint method can be derived in many different ways: - Born approimation (e.g. Tarantola) - agrange mltipliers (e.g. i & Tromp) - Data assimilation (e.g. Chen) - Operator approach elegant and strctred very general leads to simple recipes reqires a bit more mathematical machinery

11 Prelde I: Fréchet and classical derivatives The Earth model m() is a continosly defined fnction. Eamples: ρ(), v p (), v s (), Q(),. The Fréchet derivative of the misfit X(m) is the infinitesimal change of X(m) as we pass from Earth model m() to m()+δm(): d 1 m: lim dm ( m m) ( m) 2.1

12 Prelde I: Fréchet and classical derivatives The Earth model m() is a continosly defined fnction. Eamples: ρ(), v p (), v s (), Q(),. The Fréchet derivative of the misfit X(m) is the infinitesimal change of X(m) as we pass from Earth model m() to m()+δm(): d 1 m: lim dm ( m m) ( m) 2.1 For practical reasons, the continos Earth model is parameterised in terms of a finite nmber of basis fnctions b i (): N m( ) m b ( ) 2.2 i1 i i

13 Prelde I: Fréchet and classical derivatives The misfit X is then a fnction of the discrete model parameters m i : X ( m) X[ m1b 1( )... mibi ( )... mnbn ( )] 2.3 The classical partial derivatives with respect to the model parameters m i can be epressed in terms of the Fréchet derivative: d dm i 1 lim [ X ( m1b 1 1 X ( m b ( m i ) b m i b i i... m... m N N b b N N ) )] 2.4 d dm i 1 lim d dm X ( m bi ) X ( m) bi 2.5

14 Prelde I: Fréchet and classical derivatives d dm i 1 lim d dm X ( m bi ) X ( m) bi 2.5 The classical partial derivative with respect to the discrete model parameter m i is eqal to the Fréchet derivative in the direction of the basis fnction b i (). We need to find an efficient way to compte the Fréchet derivative of X for any direction (Earth model pertrbation).

15 Prelde II: Misfit fnctionals The classical (thogh not very sefl) misfit fnctional in fll waveform inversion is: 1 2 ( m; r, t) (, t) r 2 dt 2.6 synthetic waveform receiver observed waveform This can be rewritten as an integral over both time and space: ( ) dt d r The Fréchet derivative of the misfit fnctional is then: d m dm d dm r 3 ( ) dt d m, 2.8

16 Prelde II: Misfit fnctionals The Fréchet derivative of the misfit fnctional is then: m dm d r 3 ( ) dt d 2.8 In fact, the Fréchet derivative of any misfit fnctional can be written in this form: d * 3 m dm f dt d 2.9 In the special case from above: f ( ) * r 2.1 Bt f* can be anything, depending on the misfit yo choose.

17 Fréchet derivative of the misfit fnctional: Problem statement d m dm * 3 f dt d, with d m dm 2.9 The practical difficlty lies in the presence of the Fréchet derivative of the wave field, that we need to know for any model pertrbation δm. The adjoint method is a trick that allows s to eliminiate δ from eqation 2.1. The price to pay is the soltion of another differential eqation the adjoint eqation.

18 The trick d m dm * 3 f dt d, with To keep the treatment as general as possible, we write the wave eqation in a condensed form: For the special case of the 1D wave eqation: d m dm 2.9 ( ) f (, m) f (, ) m 2.11 Bt cold represent any other wave eqation (e.g. 3D anisotropic, visco-elastic, ) or even any other PDE. Compte the Fréchet derivative of in the direction δm: d(, m) d(, m) m 2.12 d dm

19 The trick d(, m) d(, m) m 2.12 d dm We can simplify the first term when is linear in (which is the case for the wave eqation): d( ) d 1 lim ( 1 lim ) ( ) ( ) ( ) ( ) ( ) 2.13 Then going back to eqation 2.13: d( ) ( ) m 2.14 dm where the eplicit dependence on m is omitted for clarity.

20 The trick d( ) ( ) m 2.14 dm Now we mltiply eqation 2.14 with an arbitrary (bt differentiable) test fnction *(,t), and integrate over time and space. d( ) 3 ( ) m dt d 2.15 dm Adding eqation 2.15 to the Fréchet derivative of the misfit d * 3 m dm f dt d 2.9 gives: d * d( ) m f ( ) m dt d dm dm

21 The trick d * d( ) m f ( ) m dt d dm dm We can eliminate δ from eqation 2.16 with the help of adjoint of : Find * and * sch that: * 3 ( ) dt d * * 3 ( ) dt d 2.17 Finding the adjoints is the actal challenge, bt if we manage to do so, we can transform eqation 2.16 to: d * ) 3 m dm d( ( m dt d dm * * 3 f ) dt d 2.19

22 The trick d * ) 3 m dm d( ( m dt d dm * * 3 f ) dt d We can eliminate δ when the adjoint field * satisfies the adjoint eqation: 2.19 * * * ( ) f 2.2 Compting the Fréchet derivative of the misfit X then becomes qite easy: d d( ) m m dt d dm dm Everything in eqation 2.21 is known.

23 Eamples The Fréchet derivative of the misfit X: d d( ) m m dt d dm dm

24 Eamples The Fréchet derivative of the misfit X: d d( ) m m dt d dm dm 1D wave eqation: ( ) ( ) 2.22 Derivative of the wave operator: d( ) d 2.23 Fréchet derivative of X: d d ( ) dt d ( ) dt d 2.24

25 Eamples The Fréchet derivative of the misfit X: d d( ) m m dt d dm dm 1D wave eqation: ( ) ( ) 2.22 Derivative of the wave operator: d( ) ( ) d 2.25 Fréchet derivatives of X: d d dt d ) ( * dt d 2.26

26 Eamples Fréchet derivatives of X: d d ( ) dt d ( ) dt d K d Fréchet kernel w.r.t. density: K dt 2.25 d d * dt d * dt d K d Fréchet kernel w.r.t. the shear modls: K * dt 2.26 Fréchet kernels are the volmetric densities of the Fréchet derivatives.

27 Eamples For the 3D elastic wave eqation: 2.27 and with a different parameterisation: 2.28 Fréchet derivatives depend on the parameterisation!!!

28 The adjoint operator for the 1D wave eqation 1D wave eqation: ( ) ( ) 3.1 with initial conditions: t t 3.2 and bondary conditions: 3.3 We need to find the adjoint field * and the adjoint operator * sch that the following eqation holds for any δ that satisfies the bondary and initial conditions: T t ( ) dt d T t ( ) dt d 3.4

29 The adjoint operator for the 1D wave eqation Find * and * sch that the following eqation holds for any δ: T t ( ) dt d T t ( ) dt d 3.4 Writing the left-hand side of 3.4 eplicitly: T t T ( ) dt d ( t ) dt d 3.5 The goal is to eliminate δ from the differentiations on the right-hand side: T t ( ) dt d d T t T t dt T t dtd 3.6

30 T t T t T t T t dtd dt d d dt ) ( The adjoint method The adjoint operator for the 1D wave eqation The goal is to eliminate δ from the differentiations on the right-hand side: 3.6 One more integration by parts: T t T t T t T t T t T t dtd dt d dt d d dt ) ( ) ( 3.7

31 T t T t T t T t dtd dt d d dt ) ( The adjoint method The adjoint operator for the 1D wave eqation The goal is to eliminate δ from the differentiations on the right-hand side: 3.6 One more integration by parts (with bondary & initial conditions acconted for): 3.8 T t T t T t T t T t dtd dt d d d dt ) ( ) (

32 The adjoint operator for the 1D wave eqation One more integration by parts (with bondary & initial conditions acconted for): 3.8 T t T t T t T t T t dtd dt d d d dt ) ( ) ( We are done, when we manage to make the bondary terms go away. So, we impose conditions pon the adjoint field * : T t T t terminal condition bondary condition (same as for )

33 The adjoint operator for the 1D wave eqation So, provided that the adjoint field * satisfies the terminal and bondary conditions tt tt terminal condition 3.9 bondary condition (same as for ) 3.1 we are left with T t ( ) dt d T t ( ) dtd 3.11 What we initially wanted to have is T t T ) dt d ( ) dt d 3.4 ( t

34 The adjoint operator for the 1D wave eqation It follows that the adjoint operator is given by ( ) ( ) 3.12 This is, fortnately, again a wave eqation, meaning that it can be solved with preeisting compter codes.

35 Smmary The Fréchet derivative of the misfit fnction: d * 3 dm The Fréchet derivative of the misfit X can be epressed in terms of the reglar wave field and the adjoint field *: d d( ) 3 m dm f dt d dm m dt d The adjoint field is governed by the adjoint eqation: * * ( ) f * The adjoint field satisfies terminal conditions: tt tt

36 Smmary The misfit can be epressed as an integral over time and space: d * 3 dm f dt d f ( ) * r 2 norm The Fréchet derivative of the misfit X can be epressed in terms of the reglar wave field and the adjoint field *: d d( ) 3 m dm dm m dt d The adjoint field is governed by the adjoint eqation: 1D wave eqation d d ( ) dt d * * ( ) f * ( ) f * 1D wave eqation The adjoint field satisfies terminal conditions: tt tt

37 The recipe 1. Solve the forward problem t 1 t 2 t 3 t 4 forward field synthetic seismograms Tape et al., 27

38 The recipe 1. Solve the forward problem t 1 t 2 t 3 t 4 forward field synthetic seismograms 2. Evalate the misfit X and compte the adjoint force 3. Solve the adjoint problem t 1 t 2 t 3 t 4 adjoint field * Tape et al., 27

39 The recipe 1. Solve the forward problem t 1 t 2 t 3 t 4 forward field synthetic seismograms 2. Evalate the misfit X and compte the adjoint force 3. Solve the adjoint problem t 1 t 2 t 3 t 4 adjoint field * 4. Compte the Fréchet kernels by integrating and * for instance: K dt Tape et al., 27

40 The recipe The interaction of the reglar and the adjoint fields generates a primary inflence zone. First-order scattering from within the primary inflence zone affects the measrement.

41 Fréchet kernel gallery measrement: cross-correlation time shift

42 Fréchet kernel gallery

43 Fréchet kernel gallery data synthetic red: Δv p > synthetics appear earlier time shift smaller ble: Δv p > synthetics appear later time shift larger

44 Fréchet kernel gallery

45 Fréchet kernel gallery

46 Practical 1. Storage of the forward field 2. Constrction of the adjoint sorce time fnction 3. Fréchet kernels

47 Practical Storage of the forward field To compte Fréchet kernels, the reglar velocity and strain fields mst be stored dring the forward simlation. Par file store velocity and strain field (1=yes, =no) every samp_ad (=1) time steps in the directory below (../INTERMEDIATE) Needed for the comptation of sensitivity kernels.

48 Practical Storage of the forward field To compte Fréchet kernels, the reglar velocity and strain fields mst be stored dring the forward simlation. Par file 2 adjoint flag = 2: read the stored forward fields and compte Fréchet kernels

49 Practical Constrction of the adjoint sorce (for cross-correlation time shifts) 1. select and isolate a waveform o & Schster, 1991 Used before in srface wave analysis.

50 Practical Constrction of the adjoint sorce (for cross-correlation time shifts) 1. select and isolate a waveform

51 Practical Constrction of the adjoint sorce (for cross-correlation time shifts) 1. select and isolate a waveform 2. compte correlation fnction C(t) (t τ )( τ)dτ τ

52 Practical Constrction of the adjoint sorce (for cross-correlation time shifts) 1. select and isolate a waveform 2. compte correlation fnction χ T 2 ΔT = cross-correlation time shift

53 Practical Constrction of the adjoint sorce (for cross-correlation time shifts) The corresponding adjoint sorce is: f r ( ( t, ) t, ) ( r ( t, ) 2 * r ) amplitde normalisation traveltime kernels are independent of the wave amplitde acts at the receiver position

54 Practical Constrction of the adjoint sorce (for cross-correlation time shifts) The corresponding adjoint sorce is: f r ( ( t, ) t, ) ( r ( t, ) 2 * r ) amplitde normalisation traveltime kernels are independent of the wave amplitde acts at the receiver position The above eqation is an approimation! It holds paradoically only when the observed and synthetic waveforms are shifted in time withot being otherwise distorted.

55 Practical Constrction of the adjoint sorce (for cross-correlation time shifts) Step 1: read synthetic seismogram and select the waveform of interest

56 Practical Constrction of the adjoint sorce (for cross-correlation time shifts) Step 2: apply a window fnction

57 Practical Constrction of the adjoint sorce (for cross-correlation time shifts) Step 3: scale and reverse in time

58 Practical Constrction of the adjoint sorce (for cross-correlation time shifts) Step 4: write the adjoint sorce time fnction and location into the following files: nmber of adjoint sorces colatitde (9 ), longitde (12.5 ), depth (1 m) ad_srcfile

59 Practical Constrction of the adjoint sorce (for cross-correlation time shifts) Step 4: write the adjoint sorce time fnction and location into the following files: ad_src_1, ad_src_2, theta-, phi- and r-component of the adjoint sorce

60 Practical Constrction of the adjoint sorce (for cross-correlation time shifts) Step 5: rn ses3d (main.ee). The otpt (Fréchet kernels) is then written to the otpt directory.

61 Practical and finally: Fréchet kernels B A A B B B A A

62 Special topics 1. The adjoint method for seismic sorce parameters 2. Time-reversal imaging of seismic sorces

63 Special topics The adjoint method for seismic sorce parameters What is the reaction of the seismic wave field to changes in the sorce parameters? ( ) f (, t) 4.1 ( ) f f (, t) (, t) 4.2 How does this change in the wave field affect the misfit X? d f??? 4.3 df

64 Special topics The adjoint method for seismic sorce parameters With the adjoint method we can answer this qestion in a straightforward way: Simply redefine the wave operator ( ) f (, m) f (, ) m 2.11 to: ( ) f (, m) m (,, f ) 4.4 Then repeat the previos derivation.

65 Special topics The adjoint method for seismic sorce parameters The adjoint eqations are eactly the same as before: The adjoint field is governed by the adjoint eqation: * * ( ) f * The adjoint field satisfies terminal conditions: tt tt Bt the Fréchet derivative (kernel) with respect to the sorce is mch simpler: d 3 f df *, t) ( f (, t) dtd * K f (, t) dt 4.5 The forward field is not involved in the Fréchet derivative and ths need not be stored.

66 Special topics The adjoint method for seismic sorce parameters Eample Misfit fnctional: ( m) dt 4.6 synthetic waveform observed waveform Adjoint sorce: f ( ) * r 4.7 Initial sorce model: f f r ( ) 4.8

67 Special topics The adjoint method for seismic sorce parameters Eample Adjoint eqation (for the 1D case): * ( ) ( * r ) 4.9 The recorded seismograms are re-injected at the sorce positions and propagated backwards in time. And the Fréchet kernel is jst the backward propagated field integrated over time: * K f (, t) dt 4.1