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Lecture 24
Exemplary Inverse Problems
including
Vibrational
Problems
Slide2Syllabus
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 BackusGilbert 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 BackusGilbert 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
Problems
Slide3Purpose of the Lecture
solve a few exemplary inverse problems
tomography
vibrational
problems
determining mean directions
Slide4Part 1
tomography
Slide5d
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
y
Slide6discretization
:
model function divided up into
M
pixels
m
j
Slide7data kernel
G
ij
= length of ray
i
in pixel
j
Slide8data kernel
G
ij
= length of ray
i
in pixel
j
here’s an
easy,
approximate way to calculate it
Slide9ray
i
start with
G
set to zero
then consider each ray in sequence
Slide10∆s
divide
each ray into segments of arc length
∆s
and step from segment to segment
Slide11determine 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)
Slide13Suppose that there are
M=L
2
voxels
A ray passes through about L
voxels
G
has
NL
2 elementsNL
of which are nonzeroso the fraction of nonzero elements is1/L
hence
G
is very sparse
Slide14In a typical
tomographic
experiment
some pixels will be missed entirely
and some groups of pixels will be sampled by only one ray
Slide15In 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
)
Slide16so
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 smooth
Slide17Solution 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 function
Slide18Solution 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 unnecessary
Slide19True model
x
y
sources and receivers
Slide20x
y
Ray Coverage
x
y
just a “few” rays shown
else image is black
Slide21Data, plotted in Radonstyle 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 gaps
Slide22Estimated model
x
y
True model
Slide23Estimated model
x
y
Estimated model
streaks due to minor data gaps
they disappear if ray density is doubled
Slide24but 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 coverage
Slide26(A)
(B)
(C)
(D)
x
y
x
y
x
y
θ
r
incomplete angular coverage
Slide27Part 2
vibrational
problems
Slide28statement of the problem
Can you determine the structure of an object
just knowing the
characteristic frequencies at which it vibrates?
frequency
Slide29the
Fréchet
derivative
of frequency with respect to velocity
is usually computed using
perturbation theory
hence a quick discussion of what that is ...
Slide30perturbation 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 known
Slide31simple example
Slide32we know the solution to this equation:
x
0
=±c
Slide34Here’s the actual
vibrational
problem
acoustic equation with
spatially variable sound velocity
v
Slide38acoustic equation with
spatially variable sound velocity
v
frequencies of vibration
or
eigenfrequencies
patterns of vibration
or
eigenfunctions
or
modes
Slide39v(
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
relationship
Slide41now represent
eigenfrequencies
and
eigenfunctions
as power series in
ε
Slide42represent
firstorder perturbed
shapes as sum of
unperturbed shapes
now represent
eigenfrequencies
and
eigenfunctions
as power series in
ε
Slide43plug series into original differential equation
group terms of equal power of
ε
solve for firstorder
perturbation
in
eigenfrequencies
ω
n
(1)and eigenfunction coefficients b
nm
(use
orthonormality in process)
Slide44result
Slide45result for
eigenfrequencies
write as standard inverse problem
Slide46standard continuous
inverse problem
Slide47standard continuous
inverse problem
perturbation in the
eigenfrequencies
are the data
perturbation in the
velocity structure is
the model function
Slide48standard continuous
inverse problem
depends upon the
unperturbed velocity structure,
the unperturbed
eigenfrequency
and the unperturbed mode
data kernel
or
Fréchet
derivative
Slide491D organ pipe
unperturbed
problem has constant velocity
0
h
x
open end,
p=0
closed end
dp
/
dx
=0
perturbed
problem has variable velocity
Slide500
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
0
Slide51solution to unperturbed problem
Slide52position , 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 structure
Slide53How to
discretize
the model function?
m
is
veloctity
function evaluated at sequence of points equally spaced in
x
our choice is very simple
Slide54the data
a list of frequencies of vibration
true, unperturbed
true, perturbed
observed = true, perturbed + noise
frequency
Slide55ω
i
m
j
the data kernel
Slide56Solution 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
Slide57Solution 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 choice
Slide58position , x
velocity, v
the solution
true
estimated
Slide59position , x
velocity, v
the solution
true
estimated
Slide60m
i
m
j
the model resolution matrix
Slide61m
i
m
j
the model resolution matrix
what is this?
Slide62This 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 end
Slide63this 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
nonuniqueness
Slide64Part 3
determining mean directions
Slide65statement of the problem
you measure a bunch of directions (unit vectors)
what’s their mean?
x
y
z
data
mean
Slide66what’s a reasonable
probability density function
for directional data?
Gaussian doesn’t quite work
because
its defined on the wrong interval
(
∞, +∞)
Slide67θ
ϕ
central vector
datum
coordinate system
distribution should
be
symmetric
in
ϕ
Slide68angle,
θ
π

π
p(
θ
)
Fisher distribution
similar in shape to a Gaussian but on a sphere
Slide69angle,
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
=1
Slide70solve by
direct application of
principle of maximum likelihood
Slide71maximize 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
Slide74Lagrange multiplier equations
Slide75Results
valid when
κ
>5
Slide76Results
central vector is parallel to the vector that you get by putting all the observed unit vectors endtoend
Slide77Solution 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
>5
Slide78Application to
Subduction
Zone Stresses
Determine the mean direction of
Paxes
of deep (300600 km) earthquakes
in the KurileKamchatka
subduction
zone
Slide79N
E
data
central direction
bootstrap
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