Wave Equation Traveltime
Inversion
Outline
• Implicit Function Theorem
• Wave Equation Traveltime Tomography
• Examples
• Generalization
• Summary
1
Implicit Function Theorem f(x,y,z)=0
Given: f(x,y,z)=0
Find: ∂z/∂x and ∂z∕∂y
∂f∕∂x
∂f∕∂y
∂z∕∂x = ; ∂z∕∂y = ∂f∕∂z
∂f∕∂z
Example: x2 + y2 + z2 = sin(xy)
S1: f(x,y,z)=x2 + y2 + z2 - sin(xy)=0
S2: ∂z∕∂y = -(2y-zcos(yz))/[2z-ycos(yz)]
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.
f=1 f=2
f(c,dT)
f=0.5
C(dT)
(dc,ddT)= dr
▼f(c,dT)
▼f(c,dT)◦dr =0 along contour
Outline
• Implicit Function Theorem
• Wave Equation Traveltime Tomography
• Examples
• Generalization
• Summary
1
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.
dT*
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*
◦
f(c,dT*)=0
f(c,dT)
C
dT(c)
dT*
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.
T=dT*
◦
∂c(x)
∂c(x)
Step 3: ∂T/∂c = - ∂f/∂c
◦
∂f/∂T
◦
◦
pred.∕∂c(x)=2G(g,t|x,0)*P(x,t|s,0)/c(x)
3.
∂P(g,t)
where
◦
◦
∂f(g,s)/∂c(x) =- ∫dt∂P(g,t)pred./∂c P(g,t+T*)obs.
◦
E = ∂f/∂T =
◦◦
SP(g,t)pred. P(g,t+T*)obs.
t
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
◦
◦
pred. P(g,t+T*)
obs.
seismograms: f(c,dT*) = 0 = SP(g,t)
t
◦
◦
Step 2: At correct lag T*, f(c,dT*) = 0 = P(g,t)pred. P(g,t)obs.
ith trace
◦
◦
◦
◦
∂fi /∂c
2/c(x)3 S [G(g,t|x,0)*P(x,t|s,0)]P(g,t+T*)obs.
Step 3: ∂T/∂c
=
=
i
i
t
i
◦
dT*
∂f/i ∂T
Ei
Migration kernel
(wavepath)
◦◦
E = SP(g,t)pred. P(g,t+T*)obs.
t
i
Step 4:
s(x)(k+1) = s(x)(k)
i
i
- a S∂Ti ∕∂c(x)[dTi]
i
Single trace
=> s(x)(k) - St [G(gi ,t|x,0)*P(x,t|s,0)]P(gi ,t+T*)obs. dTi
Traveltime weighted
WT=RTM of dt weighted trace
trace
Migration kernel
(wavepath)
Outline
• Implicit Function Theorem
• Wave Equation Traveltime Tomography
• Examples
• Generalization
• Summary
1
Fault Model
Model
WT (10 it)
WTW (14 it)
RT
1
Fault Model Data + Noise
Noisy Data
WTW (14 it)
1
Friendswood Data
Noisy Data
Median Data
Wavelet
FK Down
1
Friendswood WTW
1
Friendswood WTW vs Sonic
1
Friendswood WTW vs Sonic
1
Outline
• Implicit Function Theorem
• Wave Equation Traveltime Tomography
• Examples
• Generalization
• Summary
1
Wave Eqn Inversion Semblance Panels
1
Wave Eqn Inversion Surface Waves
1
Wave Eqn Inversion Surface Waves
1
Frequency Domain Inversion
observed
predicted
~
peak - Ai(w)peak)2
e=S(A
i(w)
i
~
e=S(wpeak - wpeak)2
i
i
i
WQT
MQA
s
z
1
Outline
• Implicit Function Theorem
• Wave Equation Traveltime Tomography
• Examples
• Generalization
• Summary
1
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
◦
◦
pred. P(g,t+T*)
obs.
seismograms: f(c,dT*) = 0 = SP(g,t)
t
◦
◦
Step 2: At correct lag T*, f(c,dT*) = 0 = P(g,t)pred. P(g,t)obs.
ith trace
◦
◦
◦
◦
∂fi /∂c
2/c(x)3 S [G(g,t|x,0)*P(x,t|s,0)]P(g,t+T*)obs.
Step 3: ∂T/∂c
=
=
i
i
t
i
◦
dT*
∂f/i ∂T
Ei
Migration kernel
(wavepath)
◦◦
E = SP(g,t)pred. P(g,t+T*)obs.
t
i
Step 4:
s(x)(k+1) = s(x)(k)
i
i
- a S∂Ti ∕∂c(x)[dTi]
i
Single trace
=> s(x)(k) - St [G(gi ,t|x,0)*P(x,t|s,0)]P(gi ,t+T*)obs. dTi
Traveltime weighted
WT=RTM of dt weighted trace
trace
Migration kernel
(wavepath)