Lecture 24
Exemplary Inverse Problems
including
Vibrational Problems
Syllabus
Lecture 01
Lecture 02
Lecture 03
Lecture 04
Lecture 05
Lecture 06
Lecture 07
Lecture 08
Lecture 09
Lecture 10
Lecture 11
Lecture 12
Lecture 13
Lecture 14
Lecture 15
Lecture 16
Lecture 17
Lecture 18
Lecture 19
Lecture 20
Lecture 21
Lecture 22
Lecture 23
Lecture 24
Describing Inverse Problems
Probability and Measurement Error, Part 1
Probability and Measurement Error, Part 2
The L2 Norm and Simple Least Squares
A Priori Information and Weighted Least Squared
Resolution and Generalized Inverses
Backus-Gilbert Inverse and the Trade Off of Resolution and Variance
The Principle of Maximum Likelihood
Inexact Theories
Nonuniqueness and Localized Averages
Vector Spaces and Singular Value Decomposition
Equality and Inequality Constraints
L1 , L∞ Norm Problems and Linear Programming
Nonlinear Problems: Grid and Monte Carlo Searches
Nonlinear Problems: Newton’s Method
Nonlinear Problems: Simulated Annealing and Bootstrap Confidence Intervals
Factor Analysis
Varimax Factors, Empircal Orthogonal Functions
Backus-Gilbert Theory for Continuous Problems; Radon’s Problem
Linear Operators and Their Adjoints
Fréchet Derivatives
Exemplary Inverse Problems, incl. Filter Design
Exemplary Inverse Problems, incl. Earthquake Location
Exemplary Inverse Problems, incl. Vibrational Problems
Purpose of the Lecture
solve a few exemplary inverse problems
tomography
vibrational problems
determining mean directions
Part 1
tomography
tomography:
data is line integral of model function
y
assume
ray path
is known
x
di = ∫ray i m(x(s), y(s)) ds
discretization:
model function divided up into M pixels mj
data kernel
Gij = length of ray i in pixel j
data kernel
Gij = length of ray i in pixel j
here’s an easy,
approximate way to
calculate it
start with G set to zero
then consider each ray in sequence
divide each ray into segments of arc length ∆s
∆s
and step from segment to segment
determine the pixel index, say j, that the center
of each line segment falls within
add ∆s to Gij
repeat for every segment of every ray
You can make this approximation
indefinitely accurate simply by
decreasing the size of ∆s
(albeit at the expense of increase the
computation time)
Suppose that there are M=L2 voxels
A ray passes through about L voxels
G has NL2 elements
NL of which are non-zero
so the fraction of non-zero elements is
1/L
hence
G is very sparse
In a typical tomographic experiment
some pixels will be missed entirely
and some groups of pixels will be sampled
by only one ray
In a typical tomographic experiment
some pixels will be missed entirely
the value of these pixels is completely undetermined
and some groups of pixels will be sampled
by only one ray
only the average value of these pixels is determined
hence the problem is mixed-determined
(and usually M>N as well)
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
smooth
Solution Possibilities
1. 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
Solution Possibilities
1. Damped Least Squares:
Matrix G is sparse and very large
use bicg() with damped least squares function
test case has very
good ray coverage,
so smoothing
probably
unnecessary
2. Weighted Least Squares:
Matrix F consists of G plus
second derivative smoothing
use bicg()with weighted least squares function
sources and
receivers
True model
trueymodel
x
0
10
10
20
20
30
30
x
x
0
40
40
50
50
60
60
0
10
20
30
y
40
50
60
0
traveltimes
0
0
just a “few”
rays shown
Ray Coverage
y with rays
true model
y
0
else image
is black
10
20
x
x
x
30
40
50
60
0
10
20
30
y
40
estimated model
50
60
Data, plotted in Radon-style
coordinates
traveltimes
0
10
10
20
20
x
distance r of ray to
center of
R image
0
30
40
30
50
40
60
-150
-100
-50
0
50
angletheta
θ of ray
100
150
Lesson from Radon’s Problem:
Full data coverage need to achieve exact solution
minor data
gaps
40
40
50
50
60
60
0
10
20
30
y
40
50
60
0
True model
10
20
30
y
40
50
60
50 50
60 60
Estimated model
traveltimes
true model
estimated
true modelmodel
with rays
0 0
0 0
10
10 10
20
20 20
10
y
20 30
30
x
x
R
x
30 30
40
40 40
50
50 50
60
60 60
40
-150
0
-100
10
-50
20
030
thetay
5040 10050 150 60
0 0
traveltimes
0
x
10 10
20 20
30 30 40 40
y y
estimated model
0
10
10
20
40
40
50
50
60
60
0
10
20
30
y
40
50
60
0
Estimated model
10
20
30
y
40
50
60
50 50
60 60
Estimated model
traveltimes
true model
estimated
true modelmodel
with rays
0 0
0 0
10
10 10
20
20 20
10
y
20 30
30
x
x
R
x
30 30
40
40 40
50
50 50
60
60 60
40
-150
0
-100
10
-50
20
030
thetay
5040 10050 150 60
0 0
traveltimes
0
10
x
10 10
20 20
30 30 40 40
y y
estimated model
streaks due to minor data gaps
10
they disappear if ray density
is doubled
0
20
but what if the observational
geometry is poor
so that broads swaths of rays are
missing ?
complete angular coverage
(A)
true model
0
10
10
20
20
y
y
30
x
x
true model with rays
(B)
0
30
40
40
50
50
60
0
10
20
60
x
30
y
(C)
40
50
60
0
10
x
20
(D)
traveltimes
40
50
60
50
60
estimated model
0
r
30
y
0
10
y
10
20
x
R
20
30
40
30
50
θ
40
x
60
-150
-100
-50
0
50
100
150
0
10
20
30
40
incomplete angular coverage
(A)
true model
0
10
10
20
20
y
x
x
y 30
30
40
40
50
50
60
60
0
10
20
(C)
30
yx
true model with rays
(B)
0
40
50
60
0
10
(D)
traveltimes
0
20
30
yx
40
50
60
50
60
estimated model
0
10
10
20
20
y 30
x
R
r
40
30
50
40
60
-150
-100
-50
0
θ
50
100
150
0
10
20
30
40
Part 2
vibrational problems
statement of the problem
Can you determine the structure of an object
just knowing the
characteristic frequencies at which it vibrates?
frequency
the Fréchet derivative
of frequency with respect to velocity
is usually computed using perturbation theory
hence a quick discussion of what that is ...
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 known
simple example
we know the
solution to this
equation: x0=±c
Here’s the actual vibrational problem
acoustic equation with
spatially variable sound velocity v
acoustic equation with
spatially variable sound velocity v
frequencies of vibration
or
eigenfrequencies
patterns of vibration
or
eigenfunctions
or
modes
assume velocity can be written as a
perturbation
around some simple structure
v(0)(x)
v(x) = v(0)(x) + εv(1)(x) + ...
eigenfunctions known to obey
orthonormality relationship
now represent eigenfrequencies and
eigenfunctions as power series in ε
now represent eigenfrequencies and
eigenfunctions as power series in ε
represent first-order
perturbed shapes as sum of
unperturbed shapes
plug series into original differential
equation
group terms of equal power of ε
solve for first-order perturbation
in eigenfrequencies ωn(1)
and eigenfunction coefficients bnm
(use orthonormality in process)
result
result for eigenfrequencies
write as standard inverse problem
standard continuous inverse problem
standard continuous inverse problem
perturbation in the
eigenfrequencies are
the data
perturbation in the
velocity structure is
the model function
standard continuous inverse problem
data kernel or Fréchet derivative
depends upon the
unperturbed velocity structure,
the unperturbed eigenfrequency
and the unperturbed mode
1D organ pipe
open end,
0
p=0
unperturbed problem has
constant velocity
perturbed problem has
variable velocity
h
closed end
x
dp/dx=0
1
0
-1
0
1
0
-1
0
0.1
0.1
0.1
0.2
0.2
0.2
0.3
0.3
0.3
0.4
0.4
0.4
0.5
0.5
0.5
0.6
0.6
0.6
0.7
0.7
0.7
0.8
0.8
0.8
0.9
0.9
0.9
1
1
1
0
𝜔1
𝜔2
𝜔3
𝜔
frequencies
x
x
x
x
1
0
dp/dx=0
h
0
p=0
0
-1
modes
p3
p2
p1
solution to unperturbed problem
10
velocity structure
8
velocity, v
6
4
2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
position , x
unperturbed
perturbed
How to discretize the model function?
our choice is very simple
m is veloctity function evaluated at sequence of
points equally spaced in x
the data
a list of frequencies of vibration
1
0.5
0
0
10
20
30
40
frequency
50
60
70
true, unperturbed
true, perturbed
observed = true, perturbed + noise
the data kernel
mj
ωi
Solution Possibilities
1. 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
Solution Possibilities
1. Damped Least Squares (implements smallness):
Matrix G is not sparse
use bicg() with damped least squares function
our choice
2. Weighted Least Squares (implements smoothness):
Matrix F consists of G plus
second derivative smoothing
use bicg()with weighted least squares function
10
the solution
8
velocity, v
6
4
2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
position , x
true
estimated
2
10
the solution
8
velocity, v
6
4
2
0
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
position , x
true
estimated
2
the model resolution matrix
mj
mi
the model resolution matrix
mj
mi
what is this?
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 end
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 nonuniqueness
Part 3
determining mean directions
statement of the problem
you measure a bunch of directions (unit vectors)
what’s their mean?
data
1
0.9
mean
0.8
0.7
3
x
z
0.6
0.5
0.4
0
0.3
0.2
0.2
0.4
0.1
0.6
0
0
0.8
0.2
0.4
0.6
0.8
x2
1
1
x1
what’s a reasonable
probability density function
for directional data?
Gaussian doesn’t quite work
because
its defined on the wrong interval
(-∞, +∞)
coordinate system
θ
ϕ
distribution should be symmetric in ϕ
Fisher distribution
similar in shape to a Gaussian but on a sphere
1
p(theta)
p(θ)
0.8
0.6
0.4
0.2
0
-3
-π
-2
-1
0
theta
angle, θ
1
2
3
π
Fisher distribution
similar in shape to a Gaussian but on a sphere
1
0.8
p(theta)
p(θ)
κ=5
0.6
0.4
κ=1
0.2
0
-3
-π
-2
-1
0
theta
angle, θ
1
2
3
π
“precision parameter”
quantifies width of p.d.f.
solve by
direct application of
principle of maximum likelihood
maximize joint p.d.f. of data
with respect to
κ and cos(θ)
x: Cartesian components of
observed unit vectors
m: Cartesian components of central
unit vector; must constrain |m|=1
likelihood function
constraint
=0
C=
unknowns
m, κ
Lagrange multiplier equations
Results
valid when κ>5
Results
central vector is parallel to
the vector that you get by
putting all the observed
unit vectors end-to-end
Solution Possibilities
Determine m by evaluating simple formula
1. Determine κ using simple but approximate formula
only valid when κ>5
2. Determine κ using bootstrap method
our choice
Application to Subduction Zone Stresses
Determine the mean direction of
P-axes
of deep (300-600 km) earthquakes
in the Kurile-Kamchatka subduction zone
N
N
1
0.8
0.6
0.4
0.2
E
0
E
-0.2
-0.4
-0.6
-0.8
-1
data
-1
central direction
-0.5
0
0.5
bootstrap
1