Toward SystemLevel Earthquake Science

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G(m)d mathematical model d data m
model G operator dG(mtrue)?
dtrue ? Forward problem find d given
m Inverse problem (discrete parameter
estimation) find
m given d Discrete linear inverse problem Gmd
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G(m)d mathematical model Discrete linear
inverse problem Gmd Method of Least
Squares Minimize E? ei2 ? (diobs-dipre)2
dipre Diobs
zi
o ei
o
o
o
o
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EeTe(d-Gm)T(d-Gm) ? di-?Gijmj
di-?Gikmk ? ? mj mk ? Gij Gik -2? mj ?
Gijdi ? di di ?/?mq ? ? mj mk ? Gij Gik
? ? ?jqmkmj?kq?GijGik
2
? mk ? Giq Gik -2 ?/?mq ? mj ? Gijdi
-2??jq?Gijdi -2? Giqdi ?/?mq ?didi0
i j k
j k i j
i i
j k i j
k i
k i
j I j i
i
i
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?/?mq 0 2 ? mk ? Giq Gik - 2? Giqdi In
matrix notation GTGm - GTd 0 mest
GTG-1GTd assuming GTG-1 exists This is the
least squares solution to Gmd
k i i
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Example of fitting a straight line mest
GTG-1GTd assuming GTG-1 exists

m ? xi
? xi ? xi2
1 1 1 1 x1
GTG 1 x2

x1 x2 .. xm .

1 xm
m ? xi -1
? xi ? xi2

GTG-1

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Example of fitting a straight line mest
GTG-1GTd assuming GTG-1 exists

? di
? xi di
1 1 1 d1
GTd d2

x1 x2 .. xm .

dm
m ? xi -1
? xi ? xi2

GTG-1 GTd

? di
? xi di
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The existence of the Least Squares
Solution mest GTG-1GTd assuming GTG-1
exists Consider the straight line problem with
only 1 data point

? ? ?

m ? xi -1 1
x1 -1 GTG-1

? xi ? xi2 x1 x12

o
The inverse of a matrix is proportional to the
reciprocal of the determinant of the matrix,
i.e., GTG-1 ? 1/(x12-x12), which is clearly
singular, and the formula for the least squares
fails.
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Classification of inverse problems Over-determin
ed Under-determined Mixed-determined Even-determin
ed


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Over-determined problems Too much information
contained in Gmd to possess an exact solution
Least squares gives a best approximate
solution.


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Even-determined problems Exactly enough
information to determine the model parameters.
There is only one solution and it has zero
prediction error


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Under-determined Problems Mixed-determined
problems - non-zero prediction error Purely
underdetermined problems - zero prediction error


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Purely Under-determined Problems of parameters
gt of equations Possible to find more than 1
solution with 0 prediction error (actually
infinitely many) To obtain a solution, we must
add some information not contained in Gmd a
priori information Example Fitting a straight
line through a single data point, we may require
that the line passes through the origin Common a
priori assumption Simple model solution best.
Measure of simplicity could be Euclidian length,
LmTm ? mi2


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Purely Under-determined Problems Problem Find
the mest that minimizes LmTm ? mi2 subject to
the constraint that ed-Gm0 ?(m) L? ?i ei ?
mi2 ? ?i di - ? Gijmj ??(m)/?mq 2 ? mi
?mi/?mq-? ?i ? Gij?mj /?mq
2mq - ? ?iGiq 0 In matrix notation 2m GT?
(1), along with Gmd
(2) Inserting (1) into (2) we get dGmGGT?/2 ,
?? 2GGT-1d and inserting into (1) m GT
GGT-1d - solution exist when purely

underdetermined


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• Mixed-determined problems
• Over Under
  Mixed
• determined determined determined
• Partition into overdetermined and underdetermined
  parts, solve by LS and minimum norm - SVD (later)
• Minimize some combination of the prediction error
  and solution length for the unpartitioned model
• ?(m)E?2LeTe?2mTm
• mestGTG?2I-1GTd - damped least squares

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• Mixed-determined problems
• ?(m)E?2LeTe?2mTm
• mestGTG?2I-1GTd - damped least squares
• Regularization parameter
• 0th-order Tikhonov Regularization

m ? Gm-d Min
m2, Gm-d2lt? min
Gm-d2, m2 lt ?
m ?
L-curves Gm-d
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Other A Priori Info Weighted Least Squares Data
weighting (weighted measures if prediction
error) EeTWee We is a weighting matrix, defining
relative contribution of each individual error to
the total prediction error (usually
diagonal). For example, for 5 observations, the
3rd may be twice as accurately determined as the
others Diag(We)1, 1, 2, 1, 1T Completely
overdetermined problem mestGTWeG-1GTWed

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Other A Priori Info Constrained
Regression dim1m2xi Constraint line must pass
through (x,d) dm1m2x Fm 1 x m1 m2T
d Similar to the unconstrained solution (2.5)
we get m1est M ? xi 1
-1 ? di m2est ? xi ? xi2
x ? xidi ????1 1
x 0 d


o
o (x,d)
o
o
d x
o
M ? xi -1
? xi ? xi2

Unconstrained solution
GTG-1 GTd

? di
? xi di
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Other A Priori Info Weighting model
parameters Instead of using minimum length as
solution simplicity, One may impose smoothness in
the model -1 1
m1 -1 1
m2 l . .
. Dm .
. .
-1 1 mN D is the flatness
matrix LlTlDmTDmmTDTDmmTWmm,
WmDTD firsth-order Tikhonov Regularization -
minGm-d22??Lm22

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Other A Priori Info Weighting model
parameters Instead of using minimum length as
solution simplicity, One may impose smoothness in
the model 1 -2 1
m1 1 -2 1
m2 l . .
. . Dm
. . . .
1 -2 1 mN D is
the roughness matrix LlTlDmTDmmTDTDmmTWmm,
WmDTD 2nd-order Tikhonov Regularization-
minGm-d22??Lm22

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