Describing Inverse Problems Syllabus Lecture 01 Describing Inverse Problems Lecture 02 Probability and Measurement Error Part 1 Lecture 03 Probability and Measurement Error Part 2 Lecture 04 The L ID: 262216
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Slide1
Lecture 1
Describing Inverse ProblemsSlide2
Syllabus
Lecture 01 Describing Inverse Problems
Lecture 02 Probability and Measurement Error, Part 1
Lecture 03 Probability and Measurement Error, Part 2
Lecture 04 The L
2
Norm and Simple Least Squares
Lecture 05 A Priori Information and Weighted Least Squared
Lecture 06 Resolution and Generalized Inverses
Lecture 07 Backus-Gilbert Inverse and the Trade Off of Resolution and Variance
Lecture 08 The Principle of Maximum Likelihood
Lecture 09 Inexact Theories
Lecture 10
Nonuniqueness
and Localized Averages
Lecture 11 Vector Spaces and Singular Value Decomposition
Lecture 12 Equality and Inequality Constraints
Lecture 13 L
1
, L
∞
Norm Problems and Linear Programming
Lecture 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 Analysis
Lecture 18
Varimax
Factors,
Empircal
Orthogonal Functions
Lecture 19 Backus-Gilbert Theory for Continuous Problems; Radon’s Problem
Lecture 20 Linear Operators and Their
Adjoints
Lecture 21
Fr
é
chet
Derivatives
Lecture 22 Exemplary Inverse Problems, incl. Filter Design
Lecture 23 Exemplary Inverse Problems, incl. Earthquake Location
Lecture 24 Exemplary Inverse Problems, incl.
Vibrational
ProblemsSlide3
Purpose of the Lecture
distinguish forward and inverse problems
categorize inverse problems
examine a few examples
enumerate different kinds of solutions to inverse problemsSlide4
Part 1
Lingo for discussing the relationship between observations and the things that we want to learn from themSlide5
three important definitionsSlide6
things that are measured in an experiment or observed in nature…
data,
d
= [
d
1
, d
2
, … dN]T
things you want to know about the world …
model parameters, m = [m1, m2, … mM]T
relationship between data and model parameters
quantitative model (or
theory
)Slide7
gravitational accelerations
travel time of seismic waves
data,
d
= [
d
1
, d
2, … dN]T
model parameters,
m = [m1, m2, … mM]T quantitative model (or
theory)
density
seismic velocity
Newton’s law of gravity
seismic wave equationSlide8
Quantitative Model
Quantitative Model
m
est
d
pre
m
est
d
obs
Forward Theory
Inverse Theory
estimates
predictions
observations
estimatesSlide9
Quantitative Model
Quantitative Model
m
true
d
pre
d
obs
≠
m
est
due to observational errorSlide10
Quantitative Model
Quantitative Model
m
true
d
pre
d
obs
≠
m
est
due to observational error
≠
due to error propagationSlide11
Understanding the effects of
observational
error
is central
to Inverse TheorySlide12
Part 2
types of quantitative models
(or
theories
)Slide13
A. Implicit Theory
L
relationships between the data and the model are knownSlide14
Example
mass = density
⨉
length
⨉ width ⨉ height
M
H
M
=
ρ
⨉
L
⨉ W ⨉ H
L
ρSlide15
weight = density
⨉
volume
measure
mass,
d
1
size,
d2,
d3, d4,want to know density, m
1
d
1
=
m
1
d
2
d
3
d
4
or
d
1
-
m
1
d
2
d
3
d
4
= 0
d
=[
d
1
,
d
2
,
d
3
,
d
4
]
T
and
N=4
m
=[
m
1
]
T
and
M=1
f
1
(
d,m
)
=0
and
L=1Slide16
note
No guarantee that
f
(
d
,
m
)=0
contains enough informationfor unique estimate mdetermining whether or not there is enoughis part of the inverse problemSlide17
B. Explicit Theory
the equation can be arranged so that
d
is a function of
m
L
=
N
one equation per datum
d = g(m) or d - g(m) = 0Slide18
Example
Circumference = 2
⨉
length
+ 2 ⨉ height
L
rectangle
H
Area = length
⨉ heightSlide19
C = 2L+2H
measure
C=
d
1
A=
d2
want to know
L=m1 H=m2
d=[
d
1
,
d
2
]
T
and
N=2
m
=[
m
1
,
m
2
]
T
and
M=2
Circumference = 2
⨉
length
+ 2 ⨉ height
Area = length
⨉ height
A=LH
d
1
= 2m
1
+ 2m
2
d
2
m
1
m
2
d=g(m)Slide20
C. Linear Explicit Theory
the function
g
(
m
) is a matrix
G
times
m
G
has N rows and M columnsd = GmSlide21
C. Linear Explicit Theory
the function
g
(
m
) is a matrix
G
times
m
G
has N rows and M columnsd = Gm
“data kernel”Slide22
Example
M
=
ρ
g
⨉
V
g
+
ρq ⨉ V qgold
quartz
total mass = density of gold
⨉
volume of gold
+ density of quartz
⨉
volume of quartz
V
= V
g
+ V
q
total volume = volume of gold
+ volume of quartzSlide23
M
=
ρ
g
⨉
V
g
+ ρq ⨉ V q
V
= V g+ V qmeasure V = d1
M =
d
2
want to know
V
g
=
m
1
V
q
=
m
2
assume
ρ
g
ρ
g
d
=[
d
1
,
d
2
]
T
and
N=2
m
=[
m
1
,
m
2
]
T
and
M=2
d
=
1
1
ρ
g
ρ
q
m
knownSlide24
D. Linear Implicit Theory
The
L
relationships between the data are linear
L
rows
N+M
columnsSlide25
in all these examples
m
is discrete
one
could have a continuous
m(x)
instead
discrete
inverse theory
continuous
inverse theorySlide26
as a discrete vector
m
in this course
we will usually approximate a continuous
m(x)
m
= [
m(
Δ
x
), m(2Δx), m(3Δx) … m(MΔ
x)]
T
but we will spend some
time later in the course dealing with the continuous problem directly Slide27
Part 3
Some ExamplesSlide28
time, t (calendar years)
temperature anomaly, T
i
(deg C)
A. Fitting a straight line to data
T = a +
btSlide29
each data point
is predicted by a straight lineSlide30
matrix formulation
d = G m
M=2Slide31
B. Fitting a parabola
T = a
+
bt
+ ct
2Slide32
each data point
is predicted by a
strquadratic
curveSlide33
matrix formulation
d = G m
M=3Slide34
straight line
note similarity
parabolaSlide35
in
MatLab
G=[ones(N,1), t, t.^2];Slide36
C. Acoustic Tomography
1
2
3
4
5
6
7
8
13
14
15
16
h
h
source,
S
receiver,
R
travel
time = length
⨉
slownessSlide37
collect data along rows and columnsSlide38
matrix formulation
d = G m
M=16
N=8Slide39
In
MatLab
G=zeros(N,M);
for
i
= [1:4]
for j = [1:4]
% measurements over rows
k = (i-1)*4 + j;
G(
i,k)=1; % measurements over columns k = (j-1)*4 + i;
G(i+4,k)=1;
end
end
Slide40
D. X-ray Imaging
S
R
1
R
2
R
3
R
4
R
5
enlarged lymph node
(A)
(B)Slide41
theory
I
= Intensity of x-rays (data)
s
= distance
c
= absorption coefficient (model
parameters)Slide42
Taylor Series
approximationSlide43
Taylor Series
approximation
discrete pixel
approximationSlide44
Taylor Series
approximation
discrete pixel
approximation
length of beam
i
in pixel j
d = G m Slide45
d = G m
matrix formulation
M
≈
10
6
N
≈
10
6Slide46
note that
G
is huge
10
6
⨉10
6
but it is sparse
(mostly zero)since a beam passes through only a tiny fraction of the total number of pixelsSlide47
in
MatLab
G =
spalloc
( N, M, MAXNONZEROELEMENTS);Slide48
E. Spectral Curve FittingSlide49
single spectral peak
area,
A
position,
f
width,
c
z
p(z)Slide50
q
spectral peaks
“
Lorentzian
”
d
=
g
(m)Slide51
e
1
e
2
e
3
e
4
e
5
e
1
e
2
e
3
e
4
e
5
s
1
s
2
ocean
sediment
F. Factor AnalysisSlide52
d
=
g
(m)Slide53
Part 4
What kind of solution are we looking for ?Slide54
A: Estimate of model parameters
meaning numerical values
m
1
= 10.5
m
2
= 7.2Slide55
But we really need confidence limits, too
m
1
= 10.5 ± 0.2
m
2
= 7.2 ± 0.1
m
1 = 10.5 ± 22.3
m2
= 7.2 ± 9.1orcompletely different implications!Slide56
B: probability density functions
if
p(m
1
)
simple
not so different than confidence intervalsSlide57
m is about 5
plus or minus 1.5
m is either
about 3
plus of minus 1
or about 8
plus or minus 1
but that’s less likely
we don’t really know anything useful about mSlide58
C: localized averages
A = 0.2m
9
+ 0.6m
10
+ 0.2m
11
might be better determined than either
m9 or m10 or
m11 individuallySlide59
Is this useful?
Do we care about
A = 0.2m
9
+ 0.6m
10
+ 0.2m
11
?Maybe …
Slide60
Suppose
if
m
is a discrete approximation of
m(x)
m(x)
x
m
10
m
11
m
9Slide61
m(x)
x
m
10
m
11
m
9
A= 0.2m
9
+ 0.6m
10
+ 0.2m
11
weighted average of
m(x)
in the vicinity of
x
10
x
10Slide62
m(x)
x
m
10
m
11
m
9
average “localized’
in the vicinity of
x
10
x
10
weights
of weighted averageSlide63
Localized average mean
can’t determine
m(x)
at
x=10
but can determine
average value of m(x) near x=10
Such a localized average might very well be useful