Lecture 23
Exemplary Inverse Problems
including
Earthquake Location
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
thermal diffusion
earthquake location
fitting of spectral peaks
Part 1
thermal diffusion
temperature in a cooling slab
h
ξ
x
erf(x)
1.0
0.5
0.0
0
1
x
2
3
temperature due to M cooling slabs
(use linear superposition)
temperature due to M slabs
each with initial temperature mj
temperature
measured at
time t>0
initial
temperature
inverse problem
infer initial temperature m
using temperatures measures at a suite of xs
at some fixed later time t
data
model
parameters
d=Gm
distance, x
0
-100
100
true temperature
0
0
3
20
2.5
40
time,
t
time
2
80
100
temperature, T
60
1.5
120
1
140
160
200
180
200
-100
0.5
-80
-60
-40
-20
0
20
distance
40
60
80
100
0
distance, x
0
-100
100
true temperature
0
0
3
initial
temperature
2.5
consists of 5
oscillations
20
40
60
time,
t
time
2
80
100
1.5
120
1
140
160
200
180
200
-100
0.5
-80
-60
-40
-20
0
20
distance
40
60
80
100
0
distance, x
0
-100
100
true temperature
0
0
3
20
2.5
40
oscillations
still
2
visible
so accurate
1.5
reconstruction
possible
time,
t
time
60
80
100
120
1
140
160
200
180
200
-100
0.5
-80
-60
-40
-20
0
20
distance
40
60
80
100
0
distance, x
0
-100
100
true temperature
0
0
3
20
2.5
40
60
time,
t
time
2
80
100
little1.5detail left,
so
1
reconstruction
will lack
resolution
0.5
120
140
160
200
180
200
-100
-80
-60
-40
-20
0
20
distance
40
60
80
100
0
What Method ?
The resolution is likely to be rather poor, especially
when data are collected at later times
damped least squares
G-g = [GTG+ε2I]-1GT
damped minimum length
G-g = GT [GGT+ε2I]-1
Backus-Gilbert
What Method ?
The resolution is likely to be rather poor, especially
when data are collected at later times
damped least squares
G-g = [GTG+ε2I]-1GT
damped minimum length
G-g = GT [GGT+ε2I]-1
Backus-Gilbert
actually, these
generalized
inverses are
equal
What Method ?
The resolution is likely to be rather poor, especially
when data are collected at later times
damped least squares
G-g = [GTG+ε2I]-1GT
damped minimum length
G-g = GT [GGT+ε2I]-1
Backus-Gilbert
might produce solutions
with fewer artifacts
Try both
damped least squares
Backus-Gilbert
Solution Possibilities
1.
Damped Least Squares:
Matrix G is not sparse
no analytic version of GTG is available
M=100 is rather small
experiment with values of ε2
mest=(G’*G+e2*eye(M,M))\(G’*d)
2.
Backus-Gilbert
use standard formulation, with damping α
experiment with values of α
Solution Possibilities
1.
Damped Least Squares:
Matrix G is not sparse
no analytic version of GTG is available
M=100 is rather small
experiment with values of ε2
mest=(G’*G+e2*eye(M,M))\(G’*d)
2.
Backus-Gilbert
use standard formulation, with damping α
experiment with values of α
try both
estimated initial temperature distribution
as a function of the time of observation
True
Damped LS
BG mest
0
0
50
50
50
150
time
time
time
100
200
-100
time
0
time
time
Backus-Gilbert
ML mest
true model
100
150
-50
0
distance
50
distance
100
200
-100
100
150
-50
0
distance
50
distance
100
200
-100
-50
0
distance
50
distance
100
estimated initial temperature distribution
as a function of the time of observation
True
Damped LS
BG mest
0
0
50
50
50
150
time
time
time
100
200
-100
time
0
time
time
Backus-Gilbert
ML mest
true model
100
150
-50
0
distance
50
distance
100
200
-100
100
150
-50
0
distance
50
distance
100
200
-100
-50
0
distance
50
distance
Damped LS
does better at
earlier times
100
estimated initial temperature distribution
as a function of the time of observation
True
Damped LS
BG mest
0
0
50
50
50
150
time
time
time
100
200
-100
time
0
time
time
Backus-Gilbert
ML mest
true model
100
150
-50
0
distance
50
distance
100
200
-100
100
150
-50
0
distance
50
distance
100
200
-100
-50
0
distance
50
distance
Damped LS
contains worse
artifacts at later
times
100
5
50
model resolution matrix when100
for
data
collected
at t=10
100
50
0
-50
-100
10
distance
Damped
LS
t=18.18
ML R at
-10
-50
-50
-50
-5
distance
distance
0
00
50
50
50
100
-100
distance
-100
-100
distance
-100
distance
distance
Backus-Gilbert
t=78.79
at t=18.18
ML RR at
BG
-50
0
distance
50
100
100
100
-100
-100
5
-50
-50
00
distance
distance
50
50
distance
distance
BG R at t=18.18
BG R at t=78.79
-100
-100
-50
-50
100
100
10
5
50
model resolution matrix when100
for
data
collected
at t=10
100
50
0
-50
-100
10
distance
Damped
LS
t=18.18
ML R at
-10
-50
-50
-50
-5
distance
distance
0
00
50
50
50
100
-100
distance
-100
-100
distance
-100
distance
distance
Backus-Gilbert
t=78.79
at t=18.18
ML RR at
BG
-50
0
distance
50
100
100
100
-100
-100
5
-50
-50
distance
BG R at t=18.18
-100
-50
00
distance
distance
50
50
distance
resolution
is similar
BG R at t=78.79
-100
-50
100
100
10
50
50
model resolution matrix when100for data collected at t=40
100
-100
-50
0
distance
50
100
-100
Damped
R LS
at t=78.79
BGML
R at
t=18.18
-50
distance
distance
distance
100100
-100
-100
-50-50
0 0
50 50
distance
distance
BG R at t=78.79
-50
50
100
0
50
distance
-100
distance
-50-50
distance
-100
50 50
100
0
distance
Backus-Gilbert
BG R at t=78.79
-100
-100
0 0
-50
100100
100
-100
-50
0
distance
distance
50
100
50
50
model resolution matrix when100for data collected at t=40
100
-100
-50
0
distance
50
100
-100
Damped
R LS
at t=78.79
BGML
R at
t=18.18
-50
distance
distance
distance
100100
-100
-100
-50-50
0 0
50 50
distance
distance
BG R at t=78.79
-50
50
100
0
50
distance
-100
distance
-50-50
distance
-100
50 50
100
0
distance
Backus-Gilbert
BG R at t=78.79
-100
-100
0 0
-50
100100
100
-100
-50
0
distance
distance
Damped LS has
much worse
sidelobes
50
100
Part 2
earthquake location
ray approximation
vibrations travel from source to receiver along
curved rays
r
x
S
s
z
P
ray approximation
vibrations travel from source to receiver along
curved rays
r
x
S wave
slower
s
S
P
z
P, S ray paths not necessarily the
same, but usually similar
P wave
faster
travel time T
integral of slowness along ray path
r
x
TS = ∫ray (1/vS) d𝓁
s
S
P
z
TP = ∫ray (1/vP) d𝓁
arrival time = travel time along ray + origin time
arrival time = travel time along ray + origin time
data
data
earthquake
location
3 model
parameters
earthquake
origin time
1 model
parameter
arrival time = travel time along ray + origin time
explicit nonlinear equation
4 model parameters
up to 2 data per station
arrival time = travel time along ray + origin time
linearize around trial
source location x(p)
tiP = TiP(x(p),x(i)) + [∇TiP] • ∆x + t0
trick is computing this gradient
Geiger’s principle
r
Δx
x(1)
x(0)
r
x(1)
Δx
x(0)
[∇TiP] = -s/v
unit vector parallel to ray pointing
away from receiver
linearized equation
Common circumstances when earthquake far from stations
All rays leave source at the same angle
x
z
All rays leave source at nearly the same angle
x
z
then, if only P wave data is available
(no S waves)
these two columns are
proportional to one-another
depth and origin time trade off
x
z
deep and late
shallow and early
Solution Possibilities
1.
2.
Damped Least Squares:
Matrix G is not sparse
no analytic version of GTG is available
M=4 is tiny
experiment with values of ε2
mest=(G’*G+e2*eye(M,M))\(G’*d)
Singular Value Decomposition
to detect case of depth and origin time trading off
test case has
earthquakes
“inside of array”
-5
3
z,xkm
0
-10
10
5
0
-5
x2
-10
-10
-8
-6
-2
-4
x1
0
2
4
6
8
10
Part 3
fitting of spectral peaks
typical spectrum consisting of
overlapping peaks
1
counts
counts
counts
0.95
0.9
0.85
xx
0.8
0.75
0.7
0.65
0
2
4
6
velocity, mm/s
8
velocity, mm/s
10
12
what shape are the peaks?
what shape are the peaks?
try both
use F test to test whether one is better than the other
what shape are the peaks?
3 unknowns
per peak
data
data
both cases:
explicit nonlinear problem
3 unknowns
per peak
linearize using analytic gradient
linearize using analytic gradient
issues
how to determine
number q of peaks
trial Ai ci fi of each peak
our solution
have operator click mouse
computer screen
to indicate position of each peak
MatLab code for graphical input
K=0;
for k = [1:20]
p = ginput(1);
if( p(1) < 0 )
break;
end
K=K+1;
a(K) = p(2)-A;
v0(K)=p(1);
c(K)=0.1;
end
1
counts
counts
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0
2
4
6
velocity, mm/s
8
velocity, mm/s
Lorentzian
Gaussian
10
12
Results of F test
Fest = E_normal/E_lorentzian: 4.230859
P(F<=1/Fest||F>=Fest) = 0.000000
Lorentzian better fit
to 99.9999% certainty