Lecture 24 Exemplary Inverse Problems including Vibrational Problems
description
Transcript of Lecture 24 Exemplary Inverse Problems including Vibrational Problems
![Page 1: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/1.jpg)
Lecture 24
Exemplary Inverse Problemsincluding
Vibrational Problems
![Page 2: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/2.jpg)
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
![Page 3: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/3.jpg)
Purpose of the Lecture
solve a few exemplary inverse problems
tomographyvibrational problems
determining mean directions
![Page 4: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/4.jpg)
Part 1
tomography
![Page 5: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/5.jpg)
di = ∫ray i m(x(s), y(s)) dsray i
tomography:data is line integral of model function
assume ray path is knownx
y
![Page 6: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/6.jpg)
discretization:model function divided up into M pixels mj
![Page 7: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/7.jpg)
data kernel
Gij = length of ray i in pixel j
![Page 8: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/8.jpg)
data kernel
Gij = length of ray i in pixel jhere’s an easy,
approximate way to calculate it
![Page 9: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/9.jpg)
ray i
start with G set to zero
then consider each ray in sequence
![Page 10: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/10.jpg)
∆s
divide each ray into segments of arc length ∆s
and step from segment to segment
![Page 11: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/11.jpg)
determine the pixel index, say j, that the center of each line segment falls within
add ∆s to Gijrepeat for every segment of every ray
![Page 12: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/12.jpg)
You can make this approximation indefinitely accurate simply by
decreasing the size of ∆s
(albeit at the expense of increase the computation time)
![Page 13: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/13.jpg)
Suppose that there are M=L2 voxels
A ray passes through about L voxels
G has NL2 elementsNL of which are non-zero
so the fraction of non-zero elements is1/Lhence G is very sparse
![Page 14: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/14.jpg)
In a typical tomographic experiment
some pixels will be missed entirely
and some groups of pixels will be sampled by only one ray
![Page 15: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/15.jpg)
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)
![Page 16: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/16.jpg)
soyou must introduce some sort of a priori
information to achieve a solutionsay
a priori information that the solution is small
or
a priori information that the solution is smooth
![Page 17: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/17.jpg)
Solution Possibilities1. 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 smoothinguse bicg()with weighted least squares function
![Page 18: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/18.jpg)
Solution Possibilities1. 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 smoothinguse bicg()with weighted least squares function
test case has very good ray coverage,
so smoothing probably
unnecessary
![Page 19: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/19.jpg)
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
xtrue model
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
true model with rays
-150 -100 -50 0 50 100 150
0
10
20
30
40
theta
R
traveltimes
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
estimated model
True model
x
y sources and receivers
![Page 20: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/20.jpg)
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
true model
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
xtrue model with rays
-150 -100 -50 0 50 100 150
0
10
20
30
40
theta
R
traveltimes
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
estimated model
x
yRay Coverage
x
y just a “few” rays shown
else image is black
![Page 21: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/21.jpg)
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
true model
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
true model with rays
-150 -100 -50 0 50 100 150
0
10
20
30
40
theta
R
traveltimes
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
estimated modelData, plotted in Radon-style coordinates
angle θ of ray
dist
ance
r of ray
to
cent
er o
f im
age
Lesson from Radon’s Problem:Full data coverage need to achieve exact solution
minor data gaps
![Page 22: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/22.jpg)
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
true model
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
true model with rays
-150 -100 -50 0 50 100 150
0
10
20
30
40
theta
R
traveltimes
0 10 20 30 40 50 60
0
10
20
30
40
50
60
yx
estimated modelEstimated model
x
y
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
true model
0 10 20 30 40 50 60
0
10
20
30
40
50
60
yx
true model with rays
-150 -100 -50 0 50 100 150
0
10
20
30
40
theta
R
traveltimes
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
estimated model
True model
![Page 23: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/23.jpg)
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
true model
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
true model with rays
-150 -100 -50 0 50 100 150
0
10
20
30
40
theta
R
traveltimes
0 10 20 30 40 50 60
0
10
20
30
40
50
60
yx
estimated modelEstimated model
x
y
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
true model
0 10 20 30 40 50 60
0
10
20
30
40
50
60
yx
true model with rays
-150 -100 -50 0 50 100 150
0
10
20
30
40
theta
R
traveltimes
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
estimated model
Estimated model
streaks due to minor data gapsthey disappear if ray density is doubled
![Page 24: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/24.jpg)
but what if the observational geometry is poor
so that broads swaths of rays are missing ?
![Page 25: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/25.jpg)
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
xtrue model
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
true model with rays
-150 -100 -50 0 50 100 150
0
10
20
30
40
theta
R
traveltimes
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
estimated model
(A) (B)
(C) (D)
x
y
x
y
x
y
θ
r
complete angular coverage
![Page 26: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/26.jpg)
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
xtrue model
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
true model with rays
-150 -100 -50 0 50 100 150
0
10
20
30
40
theta
R
traveltimes
0 10 20 30 40 50 60
0
10
20
30
40
50
60
y
x
estimated model
(A) (B)
(C) (D)
x
y
x
y
x
y
θ
r
incomplete angular coverage
![Page 27: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/27.jpg)
Part 2
vibrational problems
![Page 28: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/28.jpg)
statement of the problem
Can you determine the structure of an objectjust knowing the
characteristic frequencies at which it vibrates?
frequency
![Page 29: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/29.jpg)
the Fréchet derivativeof frequency with respect to velocity
is usually computed using perturbation theory
hence a quick discussion of what that is ...
![Page 30: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/30.jpg)
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
![Page 31: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/31.jpg)
simple example
![Page 32: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/32.jpg)
![Page 33: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/33.jpg)
we know the solution to this equation: x0=±c
![Page 34: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/34.jpg)
![Page 35: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/35.jpg)
![Page 36: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/36.jpg)
![Page 37: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/37.jpg)
Here’s the actual vibrational problem
acoustic equation withspatially variable sound velocity v
![Page 38: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/38.jpg)
acoustic equation withspatially variable sound velocity v
frequencies of vibrationor
eigenfrequencies
patterns of vibrationor
eigenfunctionsor
modes
![Page 39: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/39.jpg)
v(x) = v(0)(x) + εv(1)(x) + ...
assume velocity can be written as a perturbation
around some simple structurev(0)(x)
![Page 40: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/40.jpg)
eigenfunctions known to obey orthonormality relationship
![Page 41: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/41.jpg)
now represent eigenfrequencies and eigenfunctions as power series in ε
![Page 42: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/42.jpg)
represent first-order perturbed shapes as sum of
unperturbed shapes
now represent eigenfrequencies and eigenfunctions as power series in ε
![Page 43: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/43.jpg)
plug series into original differential equation
group terms of equal power of εsolve for first-order perturbation
in eigenfrequencies ωn(1)and eigenfunction coefficients bnm(use orthonormality in process)
![Page 44: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/44.jpg)
result
![Page 45: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/45.jpg)
result for eigenfrequencies
write as standard inverse problem
![Page 46: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/46.jpg)
standard continuous inverse problem
![Page 47: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/47.jpg)
standard continuous inverse problem
perturbation in the eigenfrequencies are the
data
perturbation in the velocity structure is the
model function
![Page 48: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/48.jpg)
standard continuous inverse problem
depends upon theunperturbed velocity structure,the unperturbed eigenfrequency
and the unperturbed mode
data kernel or Fréchet derivative
![Page 49: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/49.jpg)
1D organ pipe
unperturbed problem has constant velocity
0
h x
open end,p=0
closed enddp/dx=0perturbed problem has
variable velocity
![Page 50: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/50.jpg)
00.1
0.20.3
0.40.5
0.60.7
0.80.9
1-1 0 10
h x
p=0
dp/dx=0
p1
x
00.1
0.20.3
0.40.5
0.60.7
0.80.9
1-1 0 1
x
p2 p300.1
0.20.3
0.40.5
0.60.7
0.80.9
1-1 0 1
x𝜔1
modes
frequencies 𝜔2 𝜔3 𝜔 0
![Page 51: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/51.jpg)
solution to unperturbed problem
![Page 52: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/52.jpg)
position , x
velocity, v
perturbedunperturbed
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
6
8
10
velocity structure
![Page 53: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/53.jpg)
How to discretize the model function?
m is veloctity function evaluated at sequence of points equally spaced in x
our choice is very simple
![Page 54: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/54.jpg)
0 10 20 30 40 50 60 700
0.5
1
the dataa list of frequencies of vibration
true, unperturbedtrue, perturbedobserved = true, perturbed + noise
frequency
![Page 55: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/55.jpg)
ωi
mjthe data kernel
![Page 56: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/56.jpg)
Solution Possibilities1. 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 smoothinguse bicg()with weighted least squares function
![Page 57: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/57.jpg)
Solution Possibilities1. 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 smoothinguse bicg()with weighted least squares function
our choice
![Page 58: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/58.jpg)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
6
8
10
position , x
velocity, v
the solution
true estimated
![Page 59: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/59.jpg)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
6
8
10
position , x
velocity, v
the solution
true estimated
![Page 60: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/60.jpg)
mi
mjthe model resolution matrix
![Page 61: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/61.jpg)
mi
mjthe model resolution matrix
what is this?
![Page 62: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/62.jpg)
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
![Page 63: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/63.jpg)
this behavior is very commonand is why eigenfrequency data
are usually supplemented with other data
e.g. travel times along rays
that are not subject to this nonuniqueness
![Page 64: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/64.jpg)
Part 3
determining mean directions
![Page 65: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/65.jpg)
statement of the problemyou measure a bunch of directions (unit vectors)
what’s their mean?
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x1
x2
x 3
x y
zdata
mean
![Page 66: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/66.jpg)
what’s a reasonableprobability density function
for directional data?
Gaussian doesn’t quite workbecause
its defined on the wrong interval(-∞, +∞)
![Page 67: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/67.jpg)
θ ϕcentral vector
datum
coordinate system
distribution should be symmetric in ϕ
![Page 68: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/68.jpg)
-3 -2 -1 0 1 2 30
0.2
0.4
0.6
0.8
1
theta
p(th
eta)
angle, θ π-π
p(θ)Fisher distribution
similar in shape to a Gaussian but on a sphere
![Page 69: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/69.jpg)
-3 -2 -1 0 1 2 30
0.2
0.4
0.6
0.8
1
theta
p(th
eta)
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=1
![Page 70: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/70.jpg)
solve by
direct application of
principle of maximum likelihood
![Page 71: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/71.jpg)
maximize joint p.d.f. of data
with respect toκ and cos(θ)
![Page 72: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/72.jpg)
x: Cartesian components of observed unit vectors
m: Cartesian components of central unit vector; must constrain |m|=1
![Page 73: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/73.jpg)
likelihood function
constraint
unknownsm, κC = = 0
![Page 74: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/74.jpg)
Lagrange multiplier equations
![Page 75: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/75.jpg)
Results
valid when κ>5
![Page 76: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/76.jpg)
Results
central vector is parallel to the vector that you get by putting all the observed unit vectors end-to-end
![Page 77: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/77.jpg)
Solution PossibilitiesDetermine m by evaluating simple formula
1. Determine κ using simple but approximate formula
2. Determine κ using bootstrap method
our choice
only valid when κ>5
![Page 78: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/78.jpg)
Application to Subduction Zone Stresses
Determine the mean direction ofP-axes
of deep (300-600 km) earthquakesin the Kurile-Kamchatka subduction zone
![Page 79: Lecture 24 Exemplary Inverse Problems including Vibrational Problems](https://reader035.fdocuments.net/reader035/viewer/2022062501/5681678a550346895ddca2fb/html5/thumbnails/79.jpg)
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1N
E
N
E
data central direction bootstrap