1. Steepest Descent Optimization

2. Outline Regularized Newton Method Trust Region Method for Line Search
Solving Linear System of Equations
Traveltime Tomography
Conjugate Gradient Method

3. Problem: Ill-conditioned functional
f(x) X1 X2
Examples: 1). Many models fit the same data
2). Seismic Data with short src-rec offset insensitive to deep part of model
3). Seismic Data from shallow loud vs soft from deep part of model
4). More unknowns than equations -> non-unique solution
5). Traveltime tomography
LVZ

4. Tomography Results Tomographic Image Z (m) 400 Z (m) 400 2500 X (m)
400
Z (m)
m/s
Final smoothing operator:
50m in X , 25m in Z
400
Z (m)
Ray Path Image (Limited Offset Limits Resolution Depth)
2500
X (m)
# of Ray
Final smoothing grid size is 10x5 (50m in X , 25m in Z)

5. Seismic Refraction Data
Line 1 – Intermediate Models 1 & 2 and Final Model
(Schedule 1-3)
Intermediate
Model 1
Schedule 1
Intermediate
Model 2
Schedule 2
Final Model
Schedule 3

6. Given: f(x) ~ f(x0) + dx g + 1/2dx H dx + ½ l dx G dx
Solution: Regularized Newton
Given: f(x) ~ f(x0) + dx g + 1/2dx H dx + ½ l dx G dx
Damping parameter > 0 T T T
Misfit function Penalty function
e= (Ls-t)’(Ls-t) = t’t - s’L’t + s’L’Ls 2
Traveltime
Tomography
Gdx=Idx
Gdx ~ dx
Gdx ~ D dx 2
Find: stationary point x* s.t. f(x*)=0 D
Soln: Newton’s Method
x = x – [H + l G] g
(k) -1
(k+1)
(k) a
(k)
f(x) X1 X2
.02 max(H_ij) l
Iteration number

7. f(x) ~ f(x0) + dx g + 1/2dx H dx + ½ l dx G dx
Solution: Regularized Newton
Find: stationary point x* s.t. f(x*)=0
Soln: Newton’s Method
x = x – [H + l G] g
(k+1)
(k)
(k) -1
(k) a
.02 max(H_ij)
f(x) X1 X2 l
Iteration number
Choosing l
SD->Levenburg-Marquardt (G=I) ->Newton

8. f(x) ~ f(x0) + dx g + 1/2dx H dx + ½ l dx G dx
Solution: Regularized Newton
Find: stationary point x* s.t. f(x*)=0
Soln: Newton’s Method
x = x – [H + l G] g
(k+1)
(k)
(k) -1
(k) a
.02 max(H_ij)
f(x) X1 X2 l
Iteration number
If Hij = Hii dij then Regularized SD
x = x – g
(k+1)
(k)
(k)
[H + l G] i i i
(k) ii

9. Outline Regularized Newton Method Trust Region Method for Line Search
Solving Linear System of Equations
Traveltime Tomography
Conjugate Gradient Method

10. Soln: Let f(x) ~ -x g + 1/2x H x
Solving Square Linear Systems by SD
Given: H square matrix with SPD s.t. Hx=g
Find: x by S.D.
Soln: Let f(x) ~ -x g + 1/2x H x
Square SPD T T D
Step 1: Set f(x)=0
x = x – [H x - g]
(k+1)
(k) a
Step 2: Iterative Steepest Descent

11. Outline Regularized Newton Method Trust Region Method for Line Search
Solving Linear System of Equations
Traveltime Tomography
Conjugate Gradient Method
Rectangular &
Regularization

12. Soln: Let f(x) ~ (H x-g) (Hx-g)
Solving Rectangular Linear Systems by SD
Given: H rectangular matrix s.t. Hx=g
Find: x by S.D.
Soln: Let f(x) ~ (H x-g) (Hx-g)
Previous strategy won’t work
f(x) ~ -x’ g + 1/2x’ H x T
1/2
Step 1: Set f(x)=0 D
Step 2: Iterative Steepest Descent
Square SPD
x = x – H(H x - g)
(k+1)
(k) a T
(k)
residual

13. Soln: Let f(x) ~ (H x-g) (Hx-g) + l/2 x G x
Solving Rectangular Linear Systems by
Regularized SD
Given: H rectangular matrix s.t. Hx=g
Find: x by Regularized S.D.
Soln: Let f(x) ~ (H x-g) (Hx-g) + l/2 x G x T T
1/2
Step 1: Set f(x)=0 D
Step 2: Iterative Steepest Descent
x = x – [H(H x - g) + l Gx
(k+1)
(k) a T
(k)
(k)
Gradient or residual
Adjoint applied to residual (diff. between pred. & observed)
Migration of residual

14. Solving Rectangular Linear Systems by
Regularized SD
Given: H rectangular matrix s.t. Hx=g
x = x – [H(H x - g) + l Gx
(k+1)
(k) a T
(k) 1 5 2 x1 x2 =

15. Outline Regularized Newton Method Trust Region Method for Line Search
Solving Linear System of Equations
Traveltime Tomography
Conjugate Gradient Method
Rectangular &
Regularization&
Scaling

16. Solving Rectangular Linear Systems by Regularized SD with Scaling
Given: H rectangular matrix s.t. Hx=g ill-conditioned
Let CH H x = Cg s.t. C approximates inverse H H T T
x = x – [CH( H x - g) + l Gx
Soln:
(k+1)
(k)
(k) a T 1 5 2 x1 x2 =

17. Solving Rectangular Linear Systems by Regularized SD with Scaling
MATLAB Code
f(x) X1 X2
x = x – [CH( H x - g) + l Gx
(k+1)
(k) T
(k)

18. Outline Regularized Newton Method Trust Region Method for Line Search
Solving Linear System of Equations
Traveltime Tomography
Conjugate Gradient Method

19. Ray Based Tomography t = L s ~ s - L’dt 1. Modeling: t = Lsij j
ith ray
Problem: L is a function of s so this is a non-linear set of equations! L ij
1. Modeling: t = Ls
jth cell
2. Linearize: t = Ls subtract t’=L’s’
t-t’ = Ls-L’s’ ~ L(s-s’) dt = L ds
3. Find m that minimizes e=||t-Ls|| + l penalty 2
4. Solve: ds = [L’L] L’dt -1
5. Iterate: s = s - [L’L] L’dt -1
(k+1)
(k)
Steepest Descent
Step length
~ s - L’dt
(k) a

20. Ray Based Tomography t = L s ds ~ L dt 4. Iterate: s = s - [L’L] L’dt
ith ray
t = L s i ij j s g L ij
4. Iterate: s = s - [L’L] L’dt -1
(k+1)
(k)
~ s - L’dt
(k)
jth slowness
ds ~ L dt j ij i
Smearing residuals
that visit jth cell
Note: We never store matrix. We simply compute
a row of segment lengths (i.e., trace a ray) and then do a dot product
between that ith row vector and the column vector dt
to get the ith update to ds. Cots of each iteration is that of a matrix-vector
mulitply O(N*N) rather than O(N*N*N).
jth cell

21. Ray Based Tomography ds ~ L dt jth slowness Smearing residualsij i
Smearing residuals
that visit jth cell
jth cell
Diagonal Dominance
Preconditioning is using an approximate inverse
Regularized Steepest Descent with Preconditioning
Iterative Regularized Steepest
Descent Soln.: small memory,
no matrix inverse
Smearing weighted residuals
that visit jth cell

22. Multiscale Traveltime Inversion
# picks
Line 1 – Intermediate Models 1 & 2 and Final Model
# unknowns
(Schedule 1-3)
Coarse-grain Model
M=3N>N
Intermediate
Model 1
Schedule 1
Intermed.-grain Model
Intermediate
Model 2
Schedule 2
Finegrain Model
Final Model
Schedule 3
dx < l/4

23. Best Resolved Features Perpendicular to Ray
Note: Anomaly can be moved laterally between wells along ray and still explain data. But anomaly is restricted vertically to explain data
Where is Anomaly?
Time

24. Transmission Fresnel Zone

25. Transmission Fresnel Zone

26. Transmission Fresnel Zone
Fresnel Volume
T/2 L

27. Best Resolved Features Perpendicular to Ray{
Diffraction Ray
Snell Ray
Wavepath
Time

28. Best Resolved Features Perpendicular to Ray
Time

29. Summary t = L s ds ~ L dt = Modeling t = Ls Adjoint Modeling s = L’t jij j
Modeling
t = Ls
Note: We sum over model
space variable j
ith ray
Sum weighted slowness
Along ith ray L ij
ds ~ L dt j ij i
Adjoint Modeling
s = L’t
Note: We sum over data
space variable i rays
ith ray
Sum weighted residuals
For rays that visit jth cell

30. Outline Regularized Newton Method Trust Region Method for Line Search
Solving Linear System of Equations
Traveltime Tomography
Conjugate Gradient Method

31. Conjugate Gradient g’ = f(x*)=0 ? D dx
Quasi-Newton Condition: g’ – g = Hdx’ (1)
dx’
g’ = f(x*)=0 ? D x* g
dx’
dx’
dx’
Kiss point dx
For dx’ at the bullseye x*, g’=0 so eqn. 1 becomes, after multiplying
by dx
Conjugacy Condition: 0 = dxHdx’ (2)
x = x – a p (where p is conjugate to previous direction) (3)

32. Conjugate Gradient dx Quasi-Newton Condition: g’ – g = Hdx’ (1)
Conjugacy Condition: 0 = dxHdx’ (2)
x = x – a p (where p is conjugate to previous direction
and a linear combo of dx & g) (3)
For i = 1:nit
0 = dxH(dx + b g)
Solve for b
find b
find a
p= dx + bg x* g
dx’
dx’ = dx + ap
dx= dx’
Kiss point dx
x=x+ dx’
end