1. Wave Equation Traveltime Inversion

2. Outline Implicit Function Theorem Wave Equation Traveltime Tomography
Examples
Generalization
Summary 1

3. Implicit Function Theorem f(x,y,z)=0
Given: f(x,y,z)=0
Find: ∂z/∂x and ∂z∕∂y
∂f∕∂y
∂f∕∂x
∂f∕∂z
∂z∕∂x = -
; ∂z∕∂y = -
Example: x2 + y2 + z2 = sin(xy)
Analogous to seismic inversion problem, A is the forward modeling operator, m is the velocity model, and d is the seismogram. Because A is a nonlinear operator, the misfit function is nonlinear too and has a lot of local minima.
S1: f(x,y,z)=x2 + y2 + z2 - sin(xy)=0
S2: ∂z∕∂y = -(2y-zcos(yz))/[2z-ycos(yz)]

4. Implicit Function Theorem for f(c,dT)
This means that Frechet derivative
Of ∂data/∂model can be found even if
there isnt a PDE with data and model.
All you need is a functional equal to zero
that depends on data and model.
Analogous to seismic inversion problem, A is the forward modeling operator, m is the velocity model, and d is the seismogram. Because A is a nonlinear operator, the misfit function is nonlinear too and has a lot of local minima.
f=1
f=2
f=0.5 dT
C(dT)
f(c,dT)
▼f(c,dT)
(dc,ddT)= dr
▼f(c,dT)◦dr =0 along contour

5. Outline Implicit Function Theorem Wave Equation Traveltime Tomography
Examples
Generalization
Summary 1

6. Problem and Solution
Problem: FWI requires accurate starting model, sometimes RT tomogram not accurate enough.
Wave equation Frechet derivative (not ray-based)
Solution: Need WT tomogram: s(x)(k+1) = s(x)(k) - a S∂Ti ∕∂c(x)[∆Ti] i
However: Unlike data P and model c, which enjoy a PDE to get ∂P∕∂c we don’t have a PDE which connects data dT and model c
Step 1: Temporal correlation function between predicted and observed seismograms: f(c,dT) = P(g,t)pred. P(g,t)obs.
Step 2: At correct lag T*, f(c,dT*) = 0 = P(g,t)pred. P(g,t)obs. ◦
Analogous to seismic inversion problem, A is the forward modeling operator, m is the velocity model, and d is the seismogram. Because A is a nonlinear operator, the misfit function is nonlinear too and has a lot of local minima.
dT*

7. Problem and Solution
Problem: FWI requires accurate starting model, sometimes RT tomogram not accurate enough.
Wave equation Frechet derivative (not ray-based)
Solution: Need WT tomogram: s(x)(k+1) = s(x)(k) - a S∂Ti ∕∂c(x)[∆Ti] i
However: Unlike data P and model c, which enjoy a PDE to get ∂P∕∂c we don’t have a PDE which connects data dT and model c
Step 1: Temporal correlation function between predicted and observed seismograms: f(c,dT) = P(g,t)pred. P(g,t)obs.
Step 2: At correct lag T*, f(c,dT*) = 0 = P(g,t)pred. P(g,t)obs. ◦
T=dT*
Analogous to seismic inversion problem, A is the forward modeling operator, m is the velocity model, and d is the seismogram. Because A is a nonlinear operator, the misfit function is nonlinear too and has a lot of local minima.
f(c,dT*)=0 ◦
dT*
dT(c) C
f(c,dT)

8. Problem and Solution
Problem: FWI requires accurate starting model, sometimes RT tomogram not accurate enough.
Wave equation Frechet derivative (not ray-based)
Solution: Need WT tomogram: s(x)(k+1) = s(x)(k) - a S∂Ti ∕∂c(x)[∆Ti] i
However: Unlike data P and model c, which enjoy a PDE to get ∂P∕∂c we don’t have a PDE which connects data T and model c
Step 1: Temporal correlation function between predicted and observed seismograms: f(c,dT) = P(g,t)pred. P(g,t)obs.
Step 2: At correct lag T*, f(c,dT*) = 0 = P(g,t)pred. P(g,t)obs. ◦ ∂
∂c(x)
T=dT* ◦
∂f/∂T
Step 3: ∂T/∂c = -
∂f/∂c ◦
Analogous to seismic inversion problem, A is the forward modeling operator, m is the velocity model, and d is the seismogram. Because A is a nonlinear operator, the misfit function is nonlinear too and has a lot of local minima.
∂P(g,t)pred.∕∂c(x)=2G(g,t|x,0)*P(x,t|s,0)/c(x)3. ◦
where ◦
∂f(g,s)/∂c(x) =
- ∫dt∂P(g,t)pred./∂c P(g,t+T*)obs.
SP(g,t)pred. P(g,t+T*)obs. ◦
∂f/∂T = t
E =

9. Summary
Wave equation Frechet derivative (not ray-based)
Solution: Need WT tomogram: s(x)(k+1) = s(x)(k) - a S∂Ti ∕∂c(x)[dTi] i
Trace summation
Step 1: Connective Function: correlation of predicted and observed seismograms: f(c,dT*) = 0 = SP(g,t)pred. P(g,t+T*)obs. t ◦
Step 2: At correct lag T*, f(c,dT*) = 0 = P(g,t)pred. P(g,t)obs. ◦
2/c(x)3 S [G(g,t|x,0)*P(x,t|s,0)]P(g,t+T*)obs. ◦ i t =
∂f/ ∂T
Step 3: ∂T/∂c = -
∂f /∂c E
Migration kernel
(wavepath)
ith trace ◦
dT*
Analogous to seismic inversion problem, A is the forward modeling operator, m is the velocity model, and d is the seismogram. Because A is a nonlinear operator, the misfit function is nonlinear too and has a lot of local minima.
SP(g,t)pred. P(g,t+T*)obs. ◦
E = t i
s(x)(k+1) = s(x)(k) - a S∂Ti ∕∂c(x)[dTi]
Step 4: i
=> s(x)(k) dTi
Single trace
S [G(g ,t|x,0)*P(x,t|s,0)]P(g ,t+T*)obs. i t
Migration kernel
(wavepath)
WT=RTM of dt weighted trace
Traveltime weighted
trace

10. Outline Implicit Function Theorem Wave Equation Traveltime Tomography
Examples
Generalization
Summary 1

11. Fault Model
Model WT (10 it) WTW (14 it) RT 1

12. Fault Model Data + Noise
Noisy Data WTW (14 it) 1

13. Friendswood Data
Noisy Data Wavelet
Median Data FK Down 1

14. Friendswood WTW 1

15. Friendswood WTW vs Sonic1

16. Friendswood WTW vs Sonic1

17. Outline Implicit Function Theorem Wave Equation Traveltime Tomography
Examples
Generalization
Summary 1

18. Wave Eqn Inversion Semblance Panels1

19. Wave Eqn Inversion Surface Waves1

20. Wave Eqn Inversion Surface Waves1

21. Frequency Domain Inversion
e=S(Ai(w)peak - Ai(w)peak)2 ~ i
predicted
observed
e=S(wpeak - wpeak)2 i ~
WQT 1 s z
MQA

22. Outline Implicit Function Theorem Wave Equation Traveltime Tomography
Examples
Generalization
Summary 1

23. Summary
Wave equation Frechet derivative (not ray-based)
Solution: Need WT tomogram: s(x)(k+1) = s(x)(k) - a S∂Ti ∕∂c(x)[dTi] i
Trace summation
Step 1: Connective Function: correlation of predicted and observed seismograms: f(c,dT*) = 0 = SP(g,t)pred. P(g,t+T*)obs. t ◦
Step 2: At correct lag T*, f(c,dT*) = 0 = P(g,t)pred. P(g,t)obs. ◦
2/c(x)3 S [G(g,t|x,0)*P(x,t|s,0)]P(g,t+T*)obs. ◦ i t =
∂f/ ∂T
Step 3: ∂T/∂c = -
∂f /∂c E
Migration kernel
(wavepath)
ith trace ◦
dT*
Analogous to seismic inversion problem, A is the forward modeling operator, m is the velocity model, and d is the seismogram. Because A is a nonlinear operator, the misfit function is nonlinear too and has a lot of local minima.
SP(g,t)pred. P(g,t+T*)obs. ◦
E = t i
s(x)(k+1) = s(x)(k) - a S∂Ti ∕∂c(x)[dTi]
Step 4: i
=> s(x)(k) dTi
Single trace
S [G(g ,t|x,0)*P(x,t|s,0)]P(g ,t+T*)obs. i t
Migration kernel
(wavepath)
WT=RTM of dt weighted trace
Traveltime weighted
trace