Lecture 19 Continuous Problems: Backus-Gilbert Theory and Radon

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Lecture 19 Continuous ProblemsBackus-Gilbert
TheoryandRadons Problem
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Syllabus
Lecture 01 Describing Inverse ProblemsLecture
02 Probability and Measurement Error, Part
1Lecture 03 Probability and Measurement Error,
Part 2 Lecture 04 The L2 Norm and Simple Least
SquaresLecture 05 A Priori Information and
Weighted Least SquaredLecture 06 Resolution and
Generalized Inverses Lecture 07 Backus-Gilbert
Inverse and the Trade Off of Resolution and
VarianceLecture 08 The Principle of Maximum
LikelihoodLecture 09 Inexact TheoriesLecture
10 Nonuniqueness and Localized AveragesLecture
11 Vector Spaces and Singular Value
Decomposition Lecture 12 Equality and Inequality
ConstraintsLecture 13 L1 , L8 Norm Problems and
Linear ProgrammingLecture 14 Nonlinear
Problems Grid and Monte Carlo Searches Lecture
15 Nonlinear Problems Newtons Method Lecture
16 Nonlinear Problems Simulated Annealing and
Bootstrap Confidence Intervals Lecture
17 Factor AnalysisLecture 18 Varimax Factors,
Empircal Orthogonal FunctionsLecture
19 Backus-Gilbert Theory for Continuous Problems
Radons ProblemLecture 20 Linear Operators and
Their AdjointsLecture 21 Fréchet
DerivativesLecture 22 Exemplary Inverse
Problems, incl. Filter DesignLecture 23
Exemplary Inverse Problems, incl. Earthquake
LocationLecture 24 Exemplary Inverse Problems,
incl. Vibrational Problems
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Purpose of the Lecture
Extend Backus-Gilbert theory to continuous
problems Discuss the conversion of continuous
inverse problems to discrete problems Solve
Radons Problem the simplest tomography problem
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Part 1 Backus-Gilbert Theory
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Continuous Inverse Theorythe data are
discretebutthe model parameter is a continuous
function
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One or several dimensions
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One or several dimensions
model function
data
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hopeless to try to determine estimates of model
function at a particular depthm(z0) ?
localized average is the only way to go
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hopeless to try to determine estimates of model
function at a particular depthm(z0) ?
the problem is that an integral, such as the data
kernel integral, does not depend upon the value
of m(z) at a single point z0
localized average is the only way to go
continuous version of resolution matrix
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lets retain the idea that thesolutiondepends
linearly on the data
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lets retain the idea that thesolutiondepends
linearly on the data
continuous version of generalized inverse
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implies a formula for R
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ltmgtG-gd
comparison to discrete case
dGm
ltmgtRm
RG-gG
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implies a formula for R
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Now define the spread of resolution as
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fine generalized inversethat minimizes the
spread J with the constraint that
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J has exactly the same form as the discrete
caseonly the definition of S is different
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Hence the solution is thesame as in the discrete
case
where
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furthermore, just as we did in the discrete case,
we can add the size of the covariance
where
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as beforethis just changes the definition of S
and leads to a trade-off of resolution and
variance
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Part 2 Approximating aContinuous Problemas a
Discrete Problem
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approximation using finite number of known
functions
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approximation using finite number of known
functions
known functions
continuous function
unknown coefficients discrete model parameters
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posssible fj(x)s
voxels (and their lower dimension
equivalents) polynomials splines Fourier (and
similar) series and many others
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does the choice of fj(x) matter?
Yes! The choice implements prior
information about the properties of the
solution The solution will be different
depending upon the choice
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conversion to discrete Gmd
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special case of voxels
size controlled by the scale of variation of m(x)
integral over voxel j
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approximation when Gi(x) slowly varying
center of voxel j
size controlled by the scale of variation of Gi(x)
more stringent condition than scale of variation
of m(x)
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Part 3 Tomography
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Greek Root

• tomos
• a cut, cutting, slice, section

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tomographyas it is used in geophysics

• data are line integrals of the model function

curve i
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you can force this into the form
if you want
Gi(x)
but the Dirac delta function is not
square-integrable, which leads to problems
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Radons Problemstraight line raysdata d
treated as a continuous variable
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(u,?) coordinate system forRadon Transform
integrate over this line
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Radon Transformm(x,y) ? d(u,?)
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Inverse Problem find m(x,y) given d(u,?)
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Solution via Fourier Transforms
x?kx
kx ? x
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now Fourier transform u?ku
now change variables (u,?) ?(x,y)
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now Fourier transform u?ku
now change variables (s,u) ?(x,y)
J1, by the way
Fourier transform of d(u,?)
Fourier transform of m(x,y) evaluated on a line
of slope ?
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Learned two things

1. Proof that solution exists and unique, based on
   well-known properties of Fourier Transform
2. Recipe how to invert a Radon transform using
   Fourier transforms

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