Transcript power-point talk

Image Estimation by Example
(Claerbout’s winter class)
and its relation to Andre’s
recent tomography lecture
Large Linear Problems
Large= lots of data, Model= image
Linear= too much data for nonlinear
Why? Each survey $5M and 5 terabytes.
Subtract two surveys for fluid movement and permeability barriors.
Two-stage least squares
1. Find a differential equation that converts a
training image to random noise.
(will be the “regularization operator”)
2. Where the data fails to determine the
model, solve the differential equation
(This is “regularization”.)
Jump to movies
vector  vector
= vector
- vector
0  residual = theoretical data
- observed data
0 
m
-
d
matrix vector
-
vector
(image)
-
0 
0 
L
(5-D space)(2-D space) -
(television signal)
(3-D space)
L is an operator, not a matrix!!
• Matrix: a table of numbers
• Operator: two programs
– Prog1 = L m
– Prog2 = L’ r
(L’ is L transpose)
If you are using matrices, you are doomed!
Use Krylov subspace methods (conjugate gradients)

0 1
  
0
0
  
0 0
  
0 0
0
1
0
0
0
0
1
0
0m1 7 
   
0m2 3 

0m3 4 
   
1m4 2

0 1
  
0
0
  
0 0
  
0 0
0
1
0
0
0
0
1
0
0m1 7 
   
0m2 3 

0m3 4 
   
1m4 2
0 0
  
0
0
  
0 0
  
0 0
0
1
0
0
0
0
0
0
0m1 7 
   
0m2 3 

0m3 4 
   
1m4 2

0 0
  
0
0
  
0 0
  
0 0
0
1
0
0
0
0
0
0
0m1 7 
   
0m2 3 

0m3 4 
   
1m4 2
The null space has two members.
1
 
0
 
0
 
0
0
 
0
 
1
 
0
You can add any amount of these two
vectors to the solution m without
changing the theoretical data.
L is a matrix from physics
Example: line integral between two wells.
L usually has a null space.
0 
0 
e
L
m - d
A
m
(fitting)
(regularization)
(noise)  (tiny) (statistics)
Guess A, or find A by classical autoregression,
also called a PEF=Prediction-Error-Filter
0 L  d
   m   
0 eA n
d=data m=model (image)
L= operator from physics
A= model covariance destructor
epsilon=tiny number
n= optional noise, often taken zero
Finding the GEOFIZ solution and the GEOSTAT solution
Let n= random white noise
0 
0 
e
L
m - d
(data fitting)
A
m - n
(regularization)
GEOFIZ: Find m taking n=0
GEOSTAT: Find many m’s, one for each random noise sample
Thesis titles
• Reservoir simulation
• Reservoir uncertainty
• Reservoir estimation