lecture 23 exemplary inverse problems including earthquake location
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Lecture 23 Exemplary Inverse Problems including Earthquake Location. Syllabus. - PowerPoint PPT PresentationTRANSCRIPT
Lecture 23
Exemplary Inverse Problemsincluding
Earthquake Location
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
thermal diffusionearthquake location
fitting of spectral peaks
Part 1
thermal diffusion
xhξ0
temperature in a cooling slab
1 2 30.00.51.0
erf(x)
x
temperature due to M cooling slabs(use linear superposition)
temperature due to M slabseach 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 xsat some fixed later time t
d = G mdata model
parameters
-100 -80 -60 -40 -20 0 20 40 60 80 100
0
20
40
60
80
100
120
140
160
180
200
time
distance
true temperature
0
0.5
1
1.5
2
2.5
3
distance, x-100 100tim
e, t
0
200
0tem
perature, T
-100 -80 -60 -40 -20 0 20 40 60 80 100
0
20
40
60
80
100
120
140
160
180
200
time
distance
true temperature
0
0.5
1
1.5
2
2.5
3
distance, x-100 100tim
e, t
0
200
0initial temperature
consists of 5 oscillations
-100 -80 -60 -40 -20 0 20 40 60 80 100
0
20
40
60
80
100
120
140
160
180
200
time
distance
true temperature
0
0.5
1
1.5
2
2.5
3
distance, x-100 100tim
e, t
0
200
0oscillations still
visibleso accurate
reconstruction possible
-100 -80 -60 -40 -20 0 20 40 60 80 100
0
20
40
60
80
100
120
140
160
180
200
time
distance
true temperature
0
0.5
1
1.5
2
2.5
3
distance, x-100 100tim
e, t
0
200
0
little detail left, so
reconstruction will lack
resolution
What Method ?The resolution is likely to be rather poor, especially
when data are collected at later times
damped least squaresG-g = [GTG+ε2I]-1GT
damped minimum lengthG-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 squaresG-g = [GTG+ε2I]-1GT
damped minimum lengthG-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 squaresG-g = [GTG+ε2I]-1GT
damped minimum lengthG-g = GT [GGT+ε2I]-1
Backus-Gilbert might produce solutions with fewer artifacts
Try both
damped least squares
Backus-Gilbert
Solution Possibilities1. Damped Least Squares: Matrix G is not sparse
no analytic version of GTG is availableM=100 is rather smallexperiment with values of ε2mest=(G’*G+e2*eye(M,M))\(G’*d)
2. Backus-Gilbert use standard formulation, with damping α experiment with values of α
Solution Possibilities1. Damped Least Squares: Matrix G is not sparse
no analytic version of GTG is availableM=100 is rather smallexperiment with values of ε2mest=(G’*G+e2*eye(M,M))\(G’*d)
2. Backus-Gilbert use standard formulation, with damping α experiment with values of α
try both
-100 -50 0 50 100
0
50
100
150
200
time
distance
true model
-100 -50 0 50 100
0
50
100
150
200
time
distance
ML mest
-100 -50 0 50 100
0
50
100
150
200
time
distance
BG mest
distance distancedistance
time
time
time
True Damped LS Backus-Gilbert
estimated initial temperature distributionas a function of the time of observation
-100 -50 0 50 100
0
50
100
150
200
time
distance
true model
-100 -50 0 50 100
0
50
100
150
200
time
distance
ML mest
-100 -50 0 50 100
0
50
100
150
200
time
distance
BG mest
distance distancedistance
time
time
time
True Damped LS Backus-Gilbert
estimated initial temperature distributionas a function of the time of observation
Damped LS does better at earlier
times
-100 -50 0 50 100
0
50
100
150
200
time
distance
true model
-100 -50 0 50 100
0
50
100
150
200
time
distance
ML mest
-100 -50 0 50 100
0
50
100
150
200
time
distance
BG mest
distance distancedistance
time
time
time
True Damped LS Backus-Gilbert
estimated initial temperature distributionas a function of the time of observation
Damped LS contains worse artifacts at
later times
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
ML R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100di
stan
ce
distance
ML R at t=78.79
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
BG R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
BG R at t=78.79
Damped LS
dist
ance
distance
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
ML R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
ML R at t=78.79
-100 -50 0 50 100
-100
-50
0
50
100di
stan
ce
distance
BG R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
BG R at t=78.79
distance
Backus-Gilbert
dist
ance
model resolution matrix when for data collected at t=10
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
ML R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100di
stan
ce
distance
ML R at t=78.79
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
BG R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
BG R at t=78.79
Damped LS
dist
ance
distance
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
ML R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
ML R at t=78.79
-100 -50 0 50 100
-100
-50
0
50
100di
stan
ce
distance
BG R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
BG R at t=78.79
distance
Backus-Gilbert
dist
ance
model resolution matrix when for data collected at t=10
resolution is similar
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
ML R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
ML R at t=78.79
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
BG R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100di
stan
ce
distance
BG R at t=78.79Backus-Gilbert
distance
dist
ance
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
ML R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
ML R at t=78.79
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
BG R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
BG R at t=78.79
Damped LS
distance
dist
ance
model resolution matrix when for data collected at t=40
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
ML R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
ML R at t=78.79
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
BG R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100di
stan
ce
distance
BG R at t=78.79Backus-Gilbert
distance
dist
ance
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
ML R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
ML R at t=78.79
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
BG R at t=18.18
-100 -50 0 50 100
-100
-50
0
50
100
dist
ance
distance
BG R at t=78.79
Damped LS
distance
dist
ance
model resolution matrix when for data collected at t=40
Damped LS has much worse
sidelobes
Part 2
earthquake location
z
xP
S
s
r
ray approximationvibrations travel from source to receiver along
curved rays
z
xP
S
s
r
ray approximationvibrations travel from source to receiver along
curved rays
P wavefaster
S waveslower
P, S ray paths not necessarily the same, but usually similar
z
xP
S
s
r
TS = ∫ray (1/vS) d𝓁
travel time Tintegral of slowness along ray path
TP = ∫ray (1/vP) d𝓁
arrival time = travel time along ray + origin time
arrival time = travel time along ray + origin time
data data
earthquake location3 model
parameters
earthquake origin time
1 model parameter
arrival time = travel time along ray + origin time
explicit nonlinear equation
4 model parametersup 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
x(0)x(1) x(1) x(0)
r r
Δx Δx
rayGeiger’s principle
[∇TiP] = -s/vunit vector parallel to ray pointing
away from receiver
linearized equation
zx
zx
All rays leave source at the same angle
All rays leave source at nearly the same angle
Common circumstances when earthquake far from stations
then, if only P wave data is available
these two columns areproportional to one-another
(no S waves)
zx
depth and origin time trade off
shallow and early
deep and late
Solution Possibilities1. Damped Least Squares: Matrix G is not sparse
no analytic version of GTG is availableM=4 is tinyexperiment with values of ε2mest=(G’*G+e2*eye(M,M))\(G’*d)
2. Singular Value Decomposition to detect case of depth and origin time trading off
test case has earthquakes
“inside of array”
-10 -8 -6 -4 -2 0 2 4 6 8 10-10
-5
0
5
10
-10
-5
0
x1
x2
x 3
x, km
y, km
z, km
Part 3
fitting of spectral peaks
coun
ts
0 2 4 6 8 10 120.65
0.7
0.75
0.8
0.85
0.9
0.95
1
velocity, mm/s
coun
ts
velocity, mm/s
xx
typical spectrum consisting of overlapping peaks
coun
ts
what shape are the peaks?
what shape are the peaks?
try bothuse F test to test whether one is better than the other
what shape are the peaks?
data
3 unknownsper peak
data
3 unknownsper peak
both cases:explicit nonlinear problem
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
0 2 4 6 8 10 120.65
0.7
0.75
0.8
0.85
0.9
0.95
1
velocity, mm/s
coun
ts
velocity, mm/s
coun
ts
LorentzianGaussian
Results of F test
Fest = E_normal/E_lorentzian: 4.230859P(F<=1/Fest||F>=Fest) = 0.000000
Lorentzian better fitto 99.9999% certainty