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SyllabusLecture 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 InversesLecture 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 DecompositionLecture 12 Equality and Inequality ConstraintsLecture 13 L1 , L∞ Norm Problems and Linear ProgrammingLecture 14 Nonlinear Problems: Grid and Monte Carlo Searches Lecture 15 Nonlinear Problems: Newton’s 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; Radon’s 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
a inverse of a linear operatorcan be used to solve
a differential equation
if ℒm=f then m=ℒ-1fjust as the inverse of a matrix
can be used to solvea matrix equation
if Lm=f then m=L-1f
the adjoint can be used tomanipulate an inner product
just as the transpose can be used to manipulate the dot product
(La) Tb= a T(LTb)
(ℒa, b) =(a, ℒ†b)
first rewrite the standard inverse theory equation in terms of perturbations
a small change in the modelcauses a small change in the data
third identify the data kernel as
a kind of derivative
this kind of derivative is called aFréchet derivative
definition of a Fréchet derivative
this is mostly lingothough perhaps it adds a little insight about
what the data kernel is
,
let the data d(x) be related to the model m(x)by
could be the data kernel integral
=
because integrals are linear operators
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2
0
2
x
m(x
)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
x
d(x)
10 20 30 40 50 60 70 80 90 100-4
-2
0
iteration
log1
0 er
ror,
Em(x)
d(x) x(A)
(B)
(C)
log 10
E/E 1 xiteration
0 20 40 60 80 100 120 140 160 180 200
-0.50
0.5
x
m
0 20 40 60 80 100 120 140 160 180 200
-0.50
0.5
x
m
0 20 40 60 80 100 120 140 160 180 200-1
0
1
x
m
(A)
(B)
(C)
m(x)true
mest (x )m(1) (x
)
xxx
interpretation as tomography
backprojection
m is slownessd is travel time of a ray from –∞ to x
integrate (=add together) the travel times of all rays that pass through the point x
discrete analysis
Gm=dG= UΛVT G-g= VΛ-1UT GT= VΛUT
if Λ-1≈ Λ then G-g≈ GT backprojection works when the
singular values are all roughly the same size
suggests scalingGm=d → WGm=Wdwhere W is a diagonal matrix chosen to make the singular values more equal in overall size
Traveltime tomography:Wii = (length of ith ray)-1
so [Wd]i has interpretation of the average slowness
along the ray i.Backprojection now adds together the average slowness of all rays that
interact with the point x.
0 20 40 60
0
10
20
30
40
50
60
y
x
true model
0 20 40 60
0
10
20
30
40
50
60
y
x
LS estimate
0 20 40 60
0
10
20
30
40
50
60
y
x
BP estimate(A) (B) (C)
y
x
yx
yx
Part 4
Fréchet Derivative
involving a differential equation
seismic wave equationNavier-Stokes equation of fluid flow
etc
data d is related to field u via an inner product
field u is related to model parameters m via a differential equation
pertrubation δd is related to perturbation δu via an inner product
write in terms of perturbations
perturbation δu is related to perturbation δm via a differential equation
easy using adjoints
data is inner product with field
field satisfies ℒδu= δm
employ adjoint
inverse of adjoint is adjoint ofinverse
easy using adjoints
data is inner product with field
field satisfies ℒδu= δm
employ adjoint
inverse of adjoint is adjoint ofinverse
data kernel
easy using adjoints
data is inner product with field
field satisfies ℒδu= δm
employ adjoint
inverse of adjoint is adjoint ofinverse
data kernel
data kernel satisfies “adjoint differential equation
most problem involving differential equations are solved numerically
so instead of just solving
you must solve
and
exampletime t instead of position x
field solves a Newtonian-type heat flow equationwhere u is temperature and m is heating
data is concentration of chemical whose production rate is proportional to temperature
exampletime t instead of position x
field solves a Newtonian-type heat flow equationwhere u is temperature
data is concentration of chemical whose production rate is proportional to temperature
= (bH(ti-t), u)so hi= bH(ti-t)
we will solve this problemanalytically
using Green functions
in more complicated casesthe differential equation
must be solved numerically
note that the adjoint Green function
is the original Green function
backward in time
that’s a fairly common patternwhose significance will be pursued in a homework problem
0 50 100 150 200 250 300 350 400 450 5000
0.5
1
t
H(t
,10)
0 50 100 150 200 250 300 350 400 450 5000
1
2
t
m(t
)
0 50 100 150 200 250 300 350 400 450 5000
20
t
u(t)
0 50 100 150 200 250 300 350 400 450 5000
1000
2000
t
d(t)
(A)
(B)
(C)
(D)
time, ttime, t
time, ttime, t
F(t,τ=30
)m(t)
u(t )d(t )
previous example
unknown functi on is “forcing”
another possibility
forcing is knownparameter
is unknown
the perturbed equation is
subtracting out the unperturbed equation,ignoring second order terms, and rearranging gives ...