lecture 24 exemplary inverse problems including vibrational problems

Lecture 24

Exemplary Inverse Problemsincluding

Vibrational Problems

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

Purpose of the Lecture

solve a few exemplary inverse problems

tomographyvibrational problems

determining mean directions

Part 1

tomography

di = ∫ray i m(x(s), y(s)) dsray i

tomography:data is line integral of model function

assume ray path is knownx

y

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 jhere’s an easy,

approximate way to calculate it

ray i

start with G set to zero

then consider each ray in sequence

∆s

divide each ray into segments of arc length ∆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 Gijrepeat 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 elementsNL of which are non-zero

so the fraction of non-zero elements is1/Lhence 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

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)

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

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 smoothing

use bicg()with weighted least squares function

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 smoothing

use bicg()with weighted least squares function

test case has very good ray coverage,

so smoothing probably

unnecessary

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yRay Coverage

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else image is black

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Lesson from Radon’s Problem:Full data coverage need to achieve exact solution

minor data gaps

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Estimated model

streaks due to minor data gapsthey disappear if ray density is doubled

but what if the observational geometry is poor

so that broads swaths of rays are missing ?

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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 derivativeof 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 withspatially variable sound velocity v

acoustic equation withspatially variable sound velocity v

frequencies of vibrationor

eigenfrequencies

patterns of vibrationor

eigenfunctionsor

modes

v(x) = v(0)(x) + εv(1)(x) + ...

assume velocity can be written as a perturbation

around some simple structurev(0)(x)

eigenfunctions known to obey orthonormality relationship

now represent eigenfrequencies and eigenfunctions as power series in ε

represent first-order perturbed shapes as sum of

unperturbed shapes

now represent eigenfrequencies and eigenfunctions as power series in ε

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)

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

depends upon theunperturbed velocity structure,the unperturbed eigenfrequency

and the unperturbed mode

data kernel or Fréchet derivative

1D organ pipe

unperturbed problem has constant velocity

0

h x

open end,p=0

closed enddp/dx=0perturbed problem has

variable velocity

00.1

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0.40.5

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h x

p=0

dp/dx=0

p1

x0

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x

p2 p300.1

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x𝜔1

modes

frequencies 𝜔2 𝜔3 𝜔 0

solution to unperturbed problem

position , x

velocity, v

perturbedunperturbed

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velocity structure

How to discretize the model function?

m is veloctity function evaluated at sequence of points equally spaced in x

our choice is very simple

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the dataa list of frequencies of vibration

true, unperturbedtrue, perturbedobserved = true, perturbed + noise

frequency

ωi

mjthe data kernel

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 smoothing

use bicg()with weighted least squares function

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 smoothing

use bicg()with weighted least squares function

our choice

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position , x

velocity, v

the solution

true estimated

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the solution

true estimated

mi

mjthe model resolution matrix

mi

mjthe model resolution matrix

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 commonand 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?

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x1

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mean

what’s a reasonableprobability density function

for directional data?

Gaussian doesn’t quite workbecause

its defined on the wrong interval(-∞, +∞)

θ ϕcentral vector

datum

coordinate system

distribution should be symmetric in ϕ

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theta

p(th

eta)

angle, θ π-π

p(θ)Fisher distribution

similar in shape to a Gaussian but on a sphere

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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

solve by

direct application of

principle of maximum likelihood

maximize joint p.d.f. of data

with respect toκ and cos(θ)

x: Cartesian components of observed unit vectors

m: Cartesian components of central unit vector; must constrain |m|=1

likelihood function

constraint

unknownsm, κC = = 0

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 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

Application to Subduction Zone Stresses

Determine the mean direction ofP-axes

of deep (300-600 km) earthquakesin the Kurile-Kamchatka subduction zone

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1N

E

N

E

data central direction bootstrap