Slide 1

Report
Geology 5670/6670
Inverse Theory
17 Feb 2015
Last time: Nonlinear Inversion
• Complicating factors for solution of nonlinear
problems, i.e. of the form F(m) = d, using a standard
gradient method (i.e., Gauss-Newton) include
multiple minima and nonconvergence
• Divergence results when the sensitivity matrix G is
ill-conditioned. Stabilization of the problem can use
the generalized inverse of order p:
1 T
m k1  m k V p  p U p d k
Or a truncated Gauss step:

 T 1 T
m k 1  m k   k G k G k  G k d k


Read for Wed 18 Feb: Menke Ch 9 (163-188)
© A.R. Lowry 2015
Last time (cont’d): Nonlinear Problems
• Other approaches to stabilization of a nonlinear
problem use damping, and these include
Levenburg damping:
 T
1 T
m k 1  m k  G k G k  k I  G k d k


And Levenburg-Marquardt damping:
1
 T
 T  T
mk 1  mk  Gk Gk   kdiagGk Gk  Gk d k




Note the latter is “intermediate”, in a sense, between
Levenburg damping and a truncated Gauss step.

Assignment I is now posted on the
course website…
Due Friday, Feb 27 at the beginning of class
Criteria for stopping iteration: (Are we there yet?)
1. Reduced 2: Stop when
1
2
 
N M
N
ei2

i1
2
i
1
Note that if measurement uncertainties are overestimated,
2 for the global minimum is < 1 and final model depends
on your initial
 guess:
2 = 1 contour
m0
m0
Criteria for stopping iteration: (Are we there yet?)
2. Residual norm less than some tolerance:

d  Gm  d  Gm
2
 d  Gm 
T
3. Parameter step less than some tolerance:

m k 1  m k

mk
4. Gradient less than some tolerance:

T
E  2G d  
5. Number of iterations:

k > kmax
Solution Appraisal:
For the nonlinear problem,
1
T


C m   2 G G 


or
 T 1 1
C m  G C G 


is only approximately true, and then only when
m is “near” mtrue (and higher order derivatives are

“small”, & there are no 
other minima).
That this approach works as an approximation is because
errors arising from truncation of the Taylor series
approximation (used to derive the sensitivity matrix G
in the iterative approach) become small near mtrue.
However, generally more robust to use an analytic
confidence region…
The Likelihood Ratio
Method can be used to
generate confidence regions
from contours of E:
d  F m
N
E
2
i
i
i1
Assume measurement
errors  are jointly normal,
 zero-mean, constant
variance and uncorrelated
 E is 2 with N degrees
of freedom (if  known),
or 2 with N-M degrees
of freedom if  must be
inferred from 2 of the
minimum misfit.

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