Transcript Slide 1

Geology 5670/6670
Inverse Theory
16 Jan 2015
Last time: Course Introduction (Cont’d)
• Goals are to
 solve for “optimal” model parameters from
observational data…
 Estimate the range of models that can fit the data
within uncertainties
• We do this by minimizing the length of the residual vector,
pred
e d d
 d  G m , “assuming” that e  
N
n
• Length is defined in terms of a norm Ln = e n  n  e i
i 1
• Ordinary Least Squares 
(OLS) solves d  G m for
parameters m that minimize the L2-norm of misfit residuals

e  d  G m . (But wait… How do we do that?)
Read for Wed 21 Jan: Menke Ch 3 (39-68)

© A.R. Lowry 2015
m2
For a 2-parameter problem,
if measurement errors are
Gaussian:
Emin
m˜ 2
m1
m˜ 1

• The misfit norm E is a parabolic function of parameter
choice m

~
• “Best” estimate (here denoted m)
of mt occurs at Emin
• Contours of constant E correspond to contours of
constant confidence interval, and they describe a
probability that mt falls within the contour
So how do we find Emin?
Recall from calculus that the min or max of a function can
be found by setting the derivative = 0, so we want:
E
m1
 0;
E
m 2
0
Or equivalently:

  E 


m1
 0
E  

E



 m 2 

Recalling E = eTe, we want:
 T   T 
E  e e  2e e  0

 


Element-wise,
  


T
m1
e1
e  




 m 2 

e2
  e1

m
e N    1
  e1

 m 2
...
e2
m1
e2
m 2
...
...
and

e j
m i

G jl m l 


l1
2

m i
T
Thus:
And


 
d 
 j

e   G
 G
ji
T

 T   T

E  e e  2e

 
T

e  2G e

 e N 

m1

 e N 
 m 2 

So the solution that minimizes the L2 misfit norm is given
by:
2G
T
d  G m  0
Equivalently
T

T
G d G Gm  0
T
(MxM)
T
(Mx1)
G Gm G d
1
T
 T 
m  G G  G d



*****

1
T
 T 
Note that G G  G is called the



*****
pseudoinverse of G!
Inverse Theory: Goals include
(1) Solve for parameters from observational data;
(2) Determine the range of models that fit the data
within uncertainties
What does this tell us about uncertainty?
First, if #obs N is “large enough” relative
to M, misfit tells us something about
errors in measurements! We use that to
estimate parameter uncertainties.
Statistical Properties:
1
First denote pseudoinverse:
T

 T 
G G  G  G


And recall d  G m t   . Thus,
1
T


 T 
m˜  G G  G G m t  G   m t  G 

