including Vibrational 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: 720168
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Slide1
Lecture 24
Exemplary Inverse Problems
including
Vibrational
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
solve a few exemplary inverse problems
tomography
vibrational
problems
determining mean directionsSlide4
Part 1
tomographySlide5
d
i
= ∫
ray
i
m(x(s), y(s))
d
s
ray
i
tomography:
data is line integral of model function
assume ray path is known
x
ySlide6
discretization
:
model function divided up into
M
pixels
m
jSlide7
data kernel
G
ij
= length of ray
i
in pixel
jSlide8
data kernel
G
ij
= length of ray
i
in pixel
j
here’s an
easy,
approximate way to calculate itSlide9
ray
i
start with
G
set to zero
then consider each ray in sequenceSlide10
∆s
divide
each ray into segments of arc length
∆s
and step from segment to segment Slide11
determine the pixel
index, say
j
, that the
center
of each line segment falls within
add
∆s
to
G
ij
repeat for every segment of every ray
Slide12
You can make this approximation indefinitely accurate simply by
decreasing the size of
∆
s
(albeit at the expense of increase the computation time)Slide13
Suppose that there are
M=L
2
voxels
A ray passes through about L
voxels
G
has
NL
2 elementsNL
of which are non-zeroso the fraction of non-zero elements is1/L
hence
G
is very sparseSlide14
In a typical
tomographic
experiment
some pixels will be missed entirely
and some groups of pixels will be sampled by only one ray Slide15
In a typical
tomographic
experiment
some pixels will be missed entirely
and some groups of pixels will be sampled by only one ray
the value of these pixels is completely undetermined
only the average value of these pixels is determined
hence the problem
is mixed
-determined
(and usually
M>N
as well
)Slide16
so
you must introduce some sort of a priori information to achieve a solution
say
a priori information that the solution is small
or
a priori information that the solution is smoothSlide17
Solution Possibilities
Damped Least Squares (implements smallness):
Matrix
G
is sparse and very large
use
bicg
()
with damped least squares function
2. Weighted Least Squares (implements smoothness):
Matrix F consists of G plus second derivative smoothing use bicg
()with weighted least squares functionSlide18
Solution Possibilities
Damped Least Squares:
Matrix
G
is sparse and very large
use
bicg
()
with damped least squares function
2. Weighted Least Squares:
Matrix F consists of G plus second derivative smoothing use bicg
()with weighted least squares function
test case has very good
ray coverage, so smoothing probably unnecessarySlide19
True model
x
y
sources and receiversSlide20
x
y
Ray Coverage
x
y
just a “few” rays shown
else image is blackSlide21
Data, plotted in Radon-style coordinates
angle
θ
of ray
distance
r
of ray to center of image
Lesson from Radon’s Problem:
Full data coverage need to achieve exact solution
minor data gapsSlide22
Estimated model
x
y
True modelSlide23
Estimated model
x
y
Estimated model
streaks due to minor data gaps
they disappear if ray density is doubledSlide24
but what if the observational geometry is poor
so that broads swaths of rays are missing ?Slide25
(A)
(B)
(C)
(D)
x
y
x
y
x
y
θ
r
complete angular coverageSlide26
(A)
(B)
(C)
(D)
x
y
x
y
x
y
θ
r
incomplete angular coverageSlide27
Part 2
vibrational
problemsSlide28
statement of the problem
Can you determine the structure of an object
just knowing the
characteristic frequencies at which it vibrates?
frequencySlide29
the
Fréchet
derivative
of frequency with respect to velocity
is usually computed using
perturbation theory
hence a quick discussion of what that is ...Slide30
perturbation theory
a technique for computing an approximate solution to a complicated problem, when
1. The complicated problem is related to a simple problem by a small perturbation
2. The solution of the simple problem must be knownSlide31
simple exampleSlide32Slide33
we know the solution to this equation:
x
0
=±cSlide34Slide35Slide36Slide37
Here’s the actual
vibrational
problem
acoustic equation with
spatially variable sound velocity
vSlide38
acoustic equation with
spatially variable sound velocity
v
frequencies of vibration
or
eigenfrequencies
patterns of vibration
or
eigenfunctions
or
modesSlide39
v(
x
) = v
(0)
(
x
) +
ε
v
(1)(x) + ...assume velocity
can be written as a perturbationaround some simple structurev(0)(x)
Slide40
eigenfunctions
known to obey
orthonormality
relationshipSlide41
now represent
eigenfrequencies
and
eigenfunctions
as power series in
εSlide42
represent
first-order perturbed
shapes as sum of
unperturbed shapes
now represent
eigenfrequencies
and
eigenfunctions
as power series in
εSlide43
plug series into original differential equation
group terms of equal power of
ε
solve for first-order
perturbation
in
eigenfrequencies
ω
n
(1)and eigenfunction coefficients b
nm
(use
orthonormality in process) Slide44
resultSlide45
result for
eigenfrequencies
write as standard inverse problemSlide46
standard continuous
inverse problemSlide47
standard continuous
inverse problem
perturbation in the
eigenfrequencies
are the data
perturbation in the
velocity structure is
the model functionSlide48
standard continuous
inverse problem
depends upon the
unperturbed velocity structure,
the unperturbed
eigenfrequency
and the unperturbed mode
data kernel
or
Fréchet
derivativeSlide49
1D organ pipe
unperturbed
problem has constant velocity
0
h
x
open end,
p=0
closed end
dp
/
dx
=0
perturbed
problem has variable velocitySlide50
0
h
x
p=0
dp
/
dx
=0
p
1
x
x
p
2
p
3
x
𝜔
1
modes
frequencies
𝜔
2
𝜔
3
𝜔
0Slide51
solution to unperturbed problemSlide52
position , x
velocity, v
perturbed
unperturbed
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
0
2
4
6
8
10
velocity structureSlide53
How to
discretize
the model function?
m
is
veloctity
function evaluated at sequence of points equally spaced in
x
our choice is very simpleSlide54
the data
a list of frequencies of vibration
true, unperturbed
true, perturbed
observed = true, perturbed + noise
frequencySlide55
ω
i
m
j
the data kernelSlide56
Solution Possibilities
Damped Least Squares (implements smallness):
Matrix
G
is not sparse
use
bicg
()
with damped least squares function
2. Weighted Least Squares (implements smoothness):
Matrix F consists of G plus second derivative smoothing use bicg
()with weighted least squares functionSlide57
Solution Possibilities
Damped Least Squares (implements smallness):
Matrix
G
is not sparse
use
bicg
()
with damped least squares function
2. Weighted Least Squares (implements smoothness):
Matrix F consists of G plus second derivative smoothing use bicg
()with weighted least squares function
our choiceSlide58
position , x
velocity, v
the solution
true
estimatedSlide59
position , x
velocity, v
the solution
true
estimatedSlide60
m
i
m
j
the model resolution matrixSlide61
m
i
m
j
the model resolution matrix
what is this?Slide62
This problem has a type of
nonuniqueness
that arises from its symmetry
a positive velocity anomaly at one end of the organ pipe
trades off with a negative anomaly at the other endSlide63
this behavior is very common
and is why
eigenfrequency
data
are usually supplemented with other data
e.g. travel times along rays
that are not subject to this
nonuniquenessSlide64
Part 3
determining mean directionsSlide65
statement of the problem
you measure a bunch of directions (unit vectors)
what’s their mean?
x
y
z
data
meanSlide66
what’s a reasonable
probability density function
for directional data?
Gaussian doesn’t quite work
because
its defined on the wrong interval
(-
∞, +∞)Slide67
θ
ϕ
central vector
datum
coordinate system
distribution should
be
symmetric
in
ϕSlide68
angle,
θ
π
-
π
p(
θ
)
Fisher distribution
similar in shape to a Gaussian but on a sphereSlide69
angle,
θ
π
-
π
p(
θ
)
Fisher distribution
similar in shape to a Gaussian but on a sphere
“precision parameter”
quantifies width of
p.d.f
.
κ
=5
κ
=1Slide70
solve by
direct application of
principle of maximum likelihoodSlide71
maximize joint
p.d.f
. of data
with respect to
κ and
cos
(
θ
)
Slide72
x
: Cartesian components of observed unit vectors
m
: Cartesian components of central unit vector; must constrain
|
m
|=1
Slide73
likelihood function
constraint
unknowns
m
,
κ
C =
= 0 Slide74
Lagrange multiplier equationsSlide75
Results
valid when
κ
>5Slide76
Results
central vector is parallel to the vector that you get by putting all the observed unit vectors end-to-endSlide77
Solution Possibilities
Determine
m
by evaluating simple formula
Determine
κ
using simple but approximate formula
2. Determine
κ
using bootstrap method
our choice
only valid when
κ
>5Slide78
Application to
Subduction
Zone Stresses
Determine the mean direction of
P-axes
of deep (300-600 km) earthquakes
in the Kurile-Kamchatka
subduction
zoneSlide79
N
E
data
central direction
bootstrap