Lecture 22

Exemplary Inverse Problemsincluding

Filter Design

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

image deblurringdeconvolution filters

minimization of cross-over errors

Part 1

image deblurring

three point blur(applied to each row of pixels)

null vectors are highly oscillatory

solve with minimum length

note that GGT can deduced analytically

and is Toeplitzmight lead to a computational advantage

Solution Possibilities1. Use sparse matrix for G together with mest=G’*((G*G’)\d)

(maybe damp a little, too)

2. Use analytic version of GGTtogether with mest=G’*(GGT\d)

(maybe damp a little, too)

3. Use sparse matrix for Gtogether with bicg() to solve GGTλ=d (maybe with a little damping, too)and then use mest=GTλ

Solution Possibilities1. Use sparse matrix for G together with mest=G’*((G*G’)\d)

(maybe damp a little, too)

2. Use analytic version of GGTtogether with mest=G’*(GGT\d)

(maybe damp a little, too)

3. Use sparse matrix for Gtogether with bicg() to solve GGTλ=d (maybe with a little damping, too)and then use mest=GTλ

we used the simplest, which

worked fine

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(A) (B) (C)

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[G-g ]728

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Part 2

deconvolution filter

Convolution

general relationship forlinear systems

with translational invariance

Convolution

general relationship forlinear systems

with translational invariance

model m(t)and

data d(t)related by linear operator

Convolution

general relationship forlinear systems

with translational invariance

only relative time matters

underlying principle

linear superposition

time , t, after impulse

d(t)=

g(t)

0

time , t, after impulse

m(t)=δ(t

)0

If the input of a spike m(t)=δ(t)

spike

causes the output of d(t)=g(t)

m(t0)g(t-t0)

m(t)

time, tt0

d(t)

time, tt0

spike of amplitude, m(t0)Then the general input m(t)

causes the general output d(t)=m(t)*g(t)

convolution d=m*g

discrete convolution d=m*gstandard matrix from d=Gm

seismic reflection sounding

want airgun pulse to be as spiky as possible

p(t) = g(t) * r(t) pressure = airgun pulse * sea floor response

so as to be able to detect pulsesin sea floor responsep(t) ≈ r(t)

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0

1

time, t

x(t)

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0

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time, t

ginv (t)

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time, t

g*gin

v (t)

time, t

g(t)actual airgun pulse is ringy

so construct a deconvolution filter m(t) so that

g(t) *m(t) = δ(t) and apply it to the data

p(t)*m(t) = g(t)*m(t)*r(t) = r(t) p(t) =g(t) * r(t)

g(t) *m(t) = δ(t) and apply it to the data

p(t)*m(t) = g(t)*m(t)*r(t) = r(t) p(t) =g(t) r(t)

this is the equation we need to solve

so construct a deconvolution filter m(t) so that

100

g(t) *m(t) = δ(t) Gm = d discrete

approximation of delta function

m =

use discrete approximation of convolution

...

solve with damped least squares

mest = [GTG + ε2I]-1 GTdwith d = [1, 0, 0, ..., 0]T

(or something similar)

matrices GTG and GTd can be calculated analytically

approximately Toeplitz with elements

approximately Toeplitz with elements

autocorrelation of g

cross-correlation of g and d

Solution Possibilities1. Use sparse matrix for G together with mest=(G’*G)\(G’*d)

(maybe damping a little, too)

2. Use analytic versions of GTG and GTd together with mest=GTG\GTd

(maybe damp a little, too)

3. Never form G, just work with its columns, guse bicg() to solve GTG m = GTd but use conv() to compute GT(Gv)

4. Same as 3 but add a priori information of smoothness

Solution Possibilities1. Use sparse matrix for G together with mest=(G’*G)\(G’*d)

(maybe damping a little, too)

2. Use analytic versions of GTG and GTd together with mest=GTG\GTd

(maybe damp a little, too)

3. Never form G, just work with its columns, guse bicg() to solve GTG m = GTd but use conv() to compute GT(Gv)

4. Same as 3 but add a priori information of smoothness

we used this complicated but very fast method

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ginv (t)

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g*gin

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time, ttime, ttime, tg(t)

m(t)m(t)*

g(t)(A)

(B)

(C)

(A) Original

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d(g)

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(B) After deconvolution

d(t)d(t)*

m(t)

Part 3

minimization of cross-over errors

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tracks

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obs gravity

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longitude

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estimated

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note streaks

general ideadata s is measured along tracks

data along each track is off by an additive constant

theorysj

obs (track i) = sjtrue (track i) + m(track i)

goal is to estimate the constants by minimizing the error at track intersections

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cross-over points

ith intersection hasascending track Ai and descending track Di

sAiobs= sAi

true + mAi

sDiobs= sDi

true + mDi

subtract

sAiobs-sDi

obs= mAi- mDi

has form

d=Gm

the matrix G is very sparse

every row is all zeros, except for a single +1 and a single -1

note that this problem has an inherent non-uniqueness

m is determined only to an overall additive constant

one possibility is to use damped least squares, to choose the smallest m

(you can always add a constant later)

the matrices GTG and GTd can be calculated semi-analytically

recipestarting with zeroed GTG and GTd

Solution Possibilities1. Use sparse matrix for G together with damped least squares

mest=(G’*G+e2*speye(M,M))\(G’*d)

2. Use analytic versions of GTG and GTd add damping directly to the diagonal of GTGthen use mest=GTGpe2I\GTd

3. Use sparse matrix for Gtogether with bicg() version of damped least squares

4. Methods 1 or 2, but use hard constraint instead of damping to implement Σi mi = 0

Solution Possibilities1. Use sparse matrix for G together with damped least squares

mest=(G’*G*e2*speye(M,M))\(G’*d)

2. Use analytic versions of GTG and GTd add damping directly to the diagonal of GTGthen use mest=GTG\GTd

3. Use sparse matrix for Gtogether with bicg() version of damped least squares

4. Methods 1 or 2, but use hard constraint instead of damping

our choice

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